Excuse my ignorance but I've always been curious about this...
For example, a frog is red, but it starts living in a green forest. Over time the frog becomes green to camouflage. But a gene can't see and I'm sure there's no mechanism for color info to be transmitted to individual genes from the brain. So how does a gene know to pick green over, say, blue?
Using your example, the gene doesn't know anything. Mutations cause some of the offspring of the red frog to turn green, some to turn blue, some to turn fluorescent yellow, and some stay red. Birds can't see the green ones as well as the others, so more green frogs survive and make more green frogs. The red frogs, the fluorescent yellow ones, the blue ones, mostly get eaten. After a few generations, almost all the frogs are green -- not because the gene knew anything, not because the mutations went in any direction, but because all the other changes were counterproductive and got eaten.
The gene doesn't know anything. It's just a bunch of chemicals that randomly react with cosmic rays, chance, whatever. Most of the changes are irrelevant or actively bad, and the frog that's carrying those particular chemicals doesn't survive. But sometimes the change benefits the frog carrying the particular chemicals and then the frog sends those chemicals down to its progeny.
Obviously this is hugely over-simplified. A short and simple intro to the basics of evolution is Understanding Evolution, by UC Berkeley.
Each offspring's color is a bit different from its parents. Some colors help the frog survive, other colors tend to get it killed before it reproduces. Over time, the species tends toward a color that improves survival because those that fit their environment better will reproduce and those that fit their environment worse don't reproduce at quite the same rate.
Of course, as you know, a gene does not have any conscious, a gene does not know anything. It is all just a bunch of chemical reactions.
First, you have to understand that a gene is a piece of DNA that will be transcribed in mRNA and the mRNA will be translated into a protein (this is a bit of an oversimplification). The protein is the molecule that is causing an action. The concentration of proteins in the cell is key in causing a phenotypic effect.
The concentration of proteins can be affected by many regulatory mechanisms.
Regulation of gene expression includes a wide range of mechanisms that are used by cells to increase or decrease the production of specific gene products (protein or RNA), and is informally termed gene regulation. Sophisticated programs of gene expression are widely observed in biology, for example to trigger developmental pathways, respond to environmental stimuli, or adapt to new food sources. Virtually any step of gene expression can be modulated, from transcriptional initiation, to RNA processing, and to the post-translational modification of a protein. Often, one gene regulator controls another, and so on, in a gene regulatory network.
Post-transcriptional regulation
Post-transcriptional regulation is the control of gene expression at the RNA level, therefore between the transcription and the translation of the gene.1 It contributes substantially to gene expression regulation across human tissues.
These include mechanisms such as
Post-translational regulation refers to the control of the levels of active protein.[..] It is performed either by means of reversible events (posttranslational modifications, such as phosphorylation or sequestration) or by means of irreversible events (proteolysis).
One such regulatory mechanism must be involved.
I really don't know much about physiology and molecular biology but I could find a large number of papers (including Taylor and Hadley 1969 and Fernandez and Bagnara 1991) showing that the colour change is mediated via production of the Melanophore Stimulating Hormone (MSH) produced by the hypophysis. Maybe a better physiologist / molecular biologist could give you a better answer.
If you are interested in the particular case of color change, you might also want to have a look at Neri and Castrucci 1997 and Skold et al. 2012.
Another fun example is a certain type of moth that used to live in Western Germany ("Ruhrpott") and other areas during the Industrialisation. Its main habitat are birch trees, i.e. mainly white tree barks. So the animal used to be white with a few black spots.
Sometime in the past, the Ruhrpott had an awful lot of coal production, and in some areas the birch trees actually went and turned black from all the pollution in the air. Lo and behold, after some time, the moth also turned black with white spots.
As has been mentioned, obviously this is just basic evolution. What is nice about your and mine example is that this is a very direct, clear connection. If the animal is clearly sticking out due to its color, it will be frequently killed. It is not the other way 'round, where a slightly positive bias gets slightly more offspring, but a clear-cut, straight "be red (or white) and get caught" fact.
The effect of this is that this kind of evolution works really fast. It may have a tremendous effect after only one or two generations (obviously). So, with such drastic changes of the environment, after a handful of generations there may literally be no white moths left, the dark ones then have the whole habitat for their own (i.e., lots of food/nesting places, whatever moths do to reproduce). Hence this kind of evolution is - in contrast to other developments, which may take hundreds or thousands of years - very visible to humans.
Another fun fact: after the Industrialisation, our air turned clear, the birches turned white, and the moths got eradicated again until they were white again. Poor animals...
Source: Peppered moth evolution. Interesting read; all of this was not "obvious" to people back then, and there is criticism to be found as well.
The change in colour may be caused by new mutations or by epigenetic changes (e.g. changing the diet of a pregnant rodent may change coat color of pups, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC165709/). If the change is epigenetic, it means that the blueprint for producing the new colour was already present in the animal's genes before the colour change was triggered in its offspring by a change in the environment. In the case of the frogs, it could have been that the red frogs had some green ancestors that evolved the ability to toggle their colour between green and red in response to an environmental factor (like the availability of some specific food).
I don't know what the commenter David means with his comment "Colour adaptation does not involve changes in genes.", though. Even if the change is epigenetic, it still affects the functioning of genes.
There are actually at least three ways in which the colour of a frog might change to adapt to the colour of its environment. I will layout briefly how the three work, because from some of the comments there seems to be some confusion between the three, and also because the question mixes up changes within the lifetime of an individual frog with changes to the genome of the frog. These three categories can be more broadly applied to the question of how animals adapt to the environment.
Evolution occurs through natural selection. Other answers such as @iayork's answer have explained this mechanism.
While the genes of an organism don't change over the course of their lifetime, the way they are expressed can. This can lead to changes in skin colour based on environmental changes. This is explained in Environmental Influences on Gene Expression.
This would be a pre-existing response to a particular set of conditions; thus they 'know' what to change to because the response to the environmental change is pre-determined by their genes.
How the gene expression is influenced by the environment would depend on the particular trait in question. The mechanisms for this are not well understood, at least for some traits, as seen in The effect of light on gene expression and podophyllotoxin biosynthesis in Linum album cell culture from Plant Physiology and Biochemistry Volume 56 (July 2012, Pages 41-46)
Background adaptation is the approach used by Chameleons, but some frog species, as well as various fish and crustaceans also have this ability. In essence the distribution of pigments in specialised structures in the skin can be changed to modify the colour of the animal.
In at least some species this process relies on the animals ability of sight, so it seems likely the information on which colour to take is transmitted through the nervous system. The Wikipedia article on Chromatophores gives more information in the section on Background Adaptation
The kind of change described in the example could not be through genetic change, as this occurs over many generations, rather than one lifetime. To understand how animals adapt to the environment it is important to separate this kind of adaptation from the longer term adaptation through natural selection.
In these more rapid changes the colours to which the animal can change are already encoded in its genes. One mechanism by which the changes to the environment can be conveyed to bring about physical adaptations is through the nervous system.
DNA contains if-then statements that direct the mRNA to transcribe one piece rather than the other. I'm no microbiologist so I'm glossing here, but the transcription-mechanism is what allows proteins to be build which are then somehow assembled into hormones, tissues and various fluids such as mucus, gall, etc.
The question is really, what triggers one branch of the if-then over the other? Temperature appears to be one: deer, for example, are sexually active only in the autumn. When the temperatures drop, they start producing high-levels of sex-hormones, which cause the males to grow antlers and fight each other.
Color perception must be another: chameleons and certain species of squid are very dramatic examples. Eyes contain various cells, which are sensitive to certain wavelengths (https://en.wikipedia.org/wiki/Color_vision). The cells transmit signals to the brain, which must somehow result in the production of hormones that in turn trigger the skin-cells.
Some chemical reaches the mRNA in the relevant cells that sets the variables seasonIsAutumn or environmentIsGreen to true and the deer start growing antlers and the frogs, chameleons and squid turn green.
Of course, these signals, of temperature or color, will only have effect on those species, subspecies and genders whose DNA contains a relevant if-then statement. Deer obviously do not possess a switchColor function, just as frogs or does do not possess a growBigAntlers function.
I would like to start my answer by saying never apologize for not knowing something. There is nothing to be ashamed about and it's good to ask about things you don't know! More people should do it, and you should not be criticized for it.
I want to address your example first, since I believe there is some (common) misunderstanding. The scenario you have described is a type of Lamarckian evolution, which was actually the best attempt at explaining the mechanism of evolution before Darwinism. Lamarck was a great biologist, and so even though we may mock his ideas, it was progressive for the time with the information he had available! Anyways, Lamarck's idea was that organisms could change during their lifetime in order to adapt to their particular environment, and then pass these changes onto their offspring. The very classic example is the long neck of a giraffe. The Lamarckian thought is that a giraffe stretched his neck to reach the leaves high up in trees, and passed this newly-stretched neck onto his offspring (and hence all giraffes now have long necks). Today we know that Lamarckian evolution is never observed and is not a supported or a sufficient explanation for complex adaptations.
Here is why this idea doesn't work:
All organisms carry a one-dimensional "code" in their cell(s) which act as a recipe for "building" that organism along with the organism's characteristics that make up their phenotype (which would include the color that you describe). Most commonly this code is "written" and stored as DNA, a string of biomolecules called nucleic acids. Information is read from this code via sophisticated biological molecular machinery and used to form every aspect of the organism and that organism's effect upon the environment; that is to say the information flows from code to the environment and not the other way around (ie: it doesn't move from the environment back to the genes). I should state that there are exceptions known as epigenetics, which while extremely interesting, is beyond the scope of this question.
You can think of it this way: imagine you have a recipe for a cake. All the instructions to bake the cake are coded into a one-dimensional sequence of letters on a page of your cook book. This is the code for making the cake. You read these letters, and your brain interprets the meaning. You use this interpreted information to carry out the actions required to bake the cake. Now, once the cake is made, you cut a slice from it. Of course, the act of cutting and removing the slice does not convey information back to the recipe. You can imagine if it did, the next cake you made from that recipe would come with an already removed slice!
The point I want to get across here is that a frog which carries the gene for the red color cannot willingly or spontaneously change his color to green to match his environment. There is no feedback to the frog's genes from the environment. The ability to perform such a change would need to encoded within this frog's DNA and likely be part of a sophisticated mechanism that has evolved over millions of years. For example, many birds have breeding plumage that changes color at certain times of the year, but this is a heritable trait that has been honed through the action of natural selection over millions of years and has nothing to do with the bird's conscious desire to be more attractive.
Please do not be confused though - an organism's environment (including the "genetic environment", the collection of genes in the gene pool) as well as things like the habitat, predators, etc, are the pressures which drive evolution by natural selection. In this sense there is feedback from the environment, but it occurs within populations of species over many generations, not within individual organisms' lifetimes, and it acts through totally different means than those imagined via Lamarckian evolution.
This is a large topic and I could go on and on, but I hope that this is a sufficient answer to your specific question for now. I recommend you check out the extremely engaging and informative book called "The Blind Watchmaker" by the illustrious Richard Dawkins. My cake example was summarized from this wonderful book, and there are plenty more where that came from!
