There are several standard texts. You may want to start with a short essay, "The History of Combinatorial Game Theory", by Richard J. Nowakowski. The paper lists the main texts you may want to consult. For a more exhaustive list of references, see "Combinatorial games: Selected bibliography with a succinct GourmetIntroduction", by Aviezri Fraenkel. Thomas Ferguson, at UCLA, has an online book on game theory, and its first chapter is on combinatorial games.
The absolute minimum is "On Numbers and Games" by Conway, and "Winning Ways for your mathematical plays", by Berlekamp, Conway, and Guy. "On numbers and games" develops the general theoretical framework, so it is perhaps not what you want to look at first; "Winning ways" has many examples and treats some in large amount of detail. For completeness, let me add "Fair Game: How to Play Impartial Combinatorial Games", by Richard K. Guy.
Berlekamp has at two books on specific games, which you may want to look at if you are more interested in specific examples than the general theory: "The Dots and Boxes Game: Sophisticated Child's Play", and "Mathematical Go: Chilling Gets the Last Point".
For a more advanced treatment of some subjects, see also "Games of no chance", edited by Nowakowski. You can download several of the articles in the collection at the link. The collection includes research papers, dealing with theoretical results, but also considers specific games, connections with other fields (including economics and set theory), and includes several accessible surveys. This book is the first of a series: We also have "More games of no chance", and "Games of no chance. 3".
A very recent work that I like very much is "Combinatorial games. Tic-Tac-Toe theory", by Beck. As Beck explains, the book begins to address some of the games that the theory of "Winning games" falls a bit short of; though several examples are treated in detail, the emphasis is on the general theory.
There is also a nice recent book, "Lessons in Play: An Introduction to Combinatorial Game Theory", by Albert, Nowakowski, and Wolfe. I have seen it recommended in many places, and I agree that it is excellent.
Very recently, Aaron Siegel has developed the theory of Misere games, and he has written a book on the subject that has just appeared, "Combinatorial game theory". He is responsible for the website "Combinatorial game suite".
You may want to also look at this MO question (though it does not restrict to the combinatorial side) and to this one (which is strictly about combinatorial game theory).
Thanks to user Mark S. for suggesting that last link. By the way, Mark has a blog on precisely this topic, that also provides quite a few useful links.