So with the working definition of Eugenics: "the aim to improve the human gene pool". What are key features that distinguish Eugenics from evolution by natural selection? I mean, besides that natural selection occurs naturally.
Selection is a mechanism of evolution which favours specific forms of traits over others, this can cause the spread of beneficial mutations through a population.
Natural selection is the spread of beneficial traits/genes through populations as a result of the natural variance in their effect on reproductive output (a function of life history traits like lifespan aging and reproductive rates). A classic example is that of Darwin's finches, where different variants of bill length and shape have evolved in populations based on what food resources (types of seed) are available to them and their ability to utilise those resources.
Artificial selection is selection imposed on a population by humans to deliberately affect specific traits. Examples of this would be milk yield in cattle (we want more efficient production and better quality milk), dog breeds (selection on a whole host of traits like colour, beardedness, height, length) and crops like wheat and barley (selected for higher efficiency and quality output).
Eugenics, coming from the greek for "well-born" or "good" race, is the application of artificial selection principles to humans. Just as with cattle or crops, there is an aim, it may be to increase height, remove genetic diseases, or promote a certain set of traits like blonde hair and blue eyes. Historically we can associate with a number of radical political ideologies, such as Nazism. Even here in Sweden (people tend to think of it as a very progressive and equality driven country) there was a program of eugenics until very recently, achieved by forcibly sterilising or coercing people in to it, for various reasons (racial purity, health, and antisocial behaviour). Here's an interesting example of a girl designated as being genetically inferior because her family had a history of alcoholism (that page gives a good history of eugenics too). The social and ethical problems of eugenics are obvious.
Under normal conditions a persons reproductive output is largely determined by their own choices and actions whereas under eugenics that choice is taken away and made by someone else and specifically with the aim of "improving" the quality of the population.
So in summary:
- natural selection is the result of natural variance in fitness linked to traits
- artificial selection leads to a change in traits, in a specifically targeted direction, decided and imposed by another organism (usually humans, I know of no other species which does this)
- eugenics is practice of artificial selection on humans enacted by humans with the specific aim of improving quality: often for social, economic, or political reasons and is generally accepted as being against human rights these days.
First, you need to recognise the difference between Natural selection and Artificial selection. As a basic definition, you can say that in natural selection, selection is done on fitness (overall, long-term success of reproduction) and is determined by the complete living environment of the species. In artificial selection, selection is done by humans on a plant or animal species for specific attributes that are desired (see any Evolutionary textbook for more on the difference). So the two processes generally try to optimize completely different things, and artificial selection has a goal which is not the case for natural selection.
Generally, Eugenics can be likened to artificial selection, done by humans on humans. The methods used (selective breeding, culling, trunkated selection, sterilization etc.) are potentially the same. The difference between eugenics and articifial selection on plants and animals other than humans is purely moral and political (and therefore off-topic for Bio-SE).
However, eugenics has also often been misguided in practice (if you ignore for a moment the usually horrible morals behind it). First, it relies on desired and undesired genotypes and phenotypes. Most people would not regard these as valid concepts, and even so, what will be desired/undesired will clearly differ between societies and over time. Second, eugenics does not appreciate the value of genetic diversity for the long-term success of a population (it will rather result in lower diversity). Third, eugenics is usually aimed for obvious physical traits and diseases, without much regard for their genetic background (e.g. heterozygious recessive carriers of disease). Fourth, eugenics usually devolve into a hunt for "weaker" groups of people, even if this has nothing to do with the gene pool of the population (for instance, many birth defects and the prevalence of Down's syndrome has almost nothing to do with the genetics of the population).
I guess you could argue that ongoing selection in humans is a form of eugenics since it is to a large extent determined by human interactions and choices (see e.g. Courtiol et al 2012 & Geary et al. 2004). However, this is definitely not a standard definition, and big difference compared to "normal" eugenics is that it is not guided by a common and explicit goal. I also know that some people label the collective outcome of individual decisions as a form of eugenics, which might be relevant for how you view Preimplantation genetic diagnosis (PGD) and prenatal screening.
In principle, eugenics is just like selective breeding of animals and plants. For example, dogs have been bred for a variety of traits to fit into several niches, and corn has been bred to produce large yields in several climates. Eugenics would attempt to do the same for humans. However, no one agrees on how to "improve" the human gene pool, and far too often eugenecists have sought to remove criminals, the disabled, the jews, or whoever they feel is inferior. And the methods used to bring about their genetic utopia include forced sterilization and/or death camps. There is no moral way to implement eugenics, and I don't think humans should be in the business of deciding which people are superior or inferior and who is allowed to reproduce, it's always dangerous.