There are very goods in physics and mathematics (like Feynman lectures for exemple) that give an elementry but but broad view of their,therefore I'd like for help to find help to find a biology book that: -Covers a variety of subject in an elementary fashion. -Is written for the non-specialist.
Where I studied, every undergrad read Campbell Biology. You can start it as a non-specialist but you will not be one at the end..
The Campbell Biology is a good very introductory book to biology. It is popular book but a very standard text book. I am not aware of any popular and introductory book that are broad enough to encompass all of biology.
I would suggest that you get the campbell and in parallel you can follow free online courseware. There are lots of them. Starting with KhanAcademy is maybe a good solution. It gives some introductory courses to a bunch of disciplines. ItunesU is also a good solution to search for further introductory courses given by the best universities such as Berkeley and the MIT typically.
I would suggest something like "The Selfish Gene" by Richard Dawkins. It covers many examples or organisms and how their behaviour at a molecular level is ultimately selfish in order to ensure survival by passing their genes, which is perhaps the most important biological function of any living organism. However, as pointed in comments below, this book primarily concentrates on evolutionary biology and little else (at least in any noticeable depth) and puts forward many philosophical arguments regarding survival and passing your genes but doesn't talk about proteins or mRNA. "The Extended Phenotype" by Richard Dawkins again is an evolutionary biology book but it does talk about protein synthesis and the idea of phenotype and how genotype effects this and how the genotype of an organism can affect the environment!
Books by Richard Dawkins tends to give examples of whole organisms and in a sense the bigger picture to make his points about elementary concepts which allow life and its propagation and how it shapes the environment but doesn't go too much into details in order to keep the attention of the reader and keep it interesting to the lay audience. [for the lay audience] It is much easier to imagine a bird and its mating habits and how genes control this than knowing the molecular mechanism behind the signalling pathway responsible! However since the question is slightly subjective, my response tends to reflect a subjective response.
Alongwith campbell ,look up some "advanced" books - like Bruce Alberts molecular biology of The Cell , Lehninger, etc. It is not necessary to read everything in those books. Read whichever topics interest you the most.
There is also a very good site : ibiomagazine.org which has videos of some of the big shots of biology.
And remember, most important is to keep all your senses open always. There is so much biology going around. Observing and thinking about it can teach you things which 100 textbooks can't !