I'm looking for a book with the description of basic types of graphs, terminology used in this field of Mathematics and main theorems. All in all, a good book to start with to be able to understand other more complicated works.
Diestel's book is not exactly light reading but it's thorough, current and really good. Also in the GTM series is Bollobas' book which is very good as well, and covers somewhat different ground with a different angle (Diestel emphasizes the forcing relationships between various invariants which is a nice unifying theme).
There are hundreds of other introductory texts, but I would go with one of these two (or both).

1$\begingroup$ Diestel's book Graph Theory is also freely available to view online. diestelgraphtheory.com $\endgroup$ – András Salamon Sep 16 '10 at 10:40

Introduction to Graph Theory by D.B. West. The book homepage is maintained by the author. Well written, with plenty of exercises.

$\begingroup$ I was starting to think nobody would mention it! $\endgroup$ – Andrew D. King Sep 16 '10 at 16:06
1) Harary's book is great  and he's a hoot.
2) Ringel and Hartsfield's Pearl in Graph Theory is great, lovable and has lots of pictures and excellent exercises  ideal for an undergrad class that's not geared towards prepping students towards a grad course.
3) Gross and Yellen's book is probably the best introduction if you're a "mature" math reader and want to really learn stuff  this is what I would buy.
4) Applied Combinatorics by Tucker is phenomenal  but graph theory is only 4 chapters out of 11. Nonetheless, I'd strongly consider this one.

2$\begingroup$ Ringel and Hartsfield's Pearls in Graph Theory is indeed a gem. $\endgroup$ – Felix Goldberg Apr 29 '12 at 23:48

3$\begingroup$ There is also "Extra pearls in graph theory" arxiv.org/abs/1812.06627, it is a supplement to "Pearls in graph theory" covering a few additional topics. $\endgroup$ – Dmitri Panov Dec 27 '18 at 10:55
Robin Wilson's Introduction to Graph Theory is very easy to read  I read it over a weekend. I definitely recommend you give this a quick read before plunging into Bondy and Murty, Diestel or West.
The classic Graph Theory with Applications by Bondy and Murty is available online.

$\begingroup$ I would recommend the new edition of their book also. $\endgroup$ – Emil Nov 10 '09 at 15:24
Graphs & digraphs by Gary Chartrand and Linda Lesniak is very well written. The organization is nice and the proofs are very clear. Moreover the exercises are concrete and to the point.
Pearls in Graph Theory: A Comprehensive Introduction by Nora Hartsfield and Gerhard Ringel. This is a great introductory book and is one of the better dover books out there in my opinion.

$\begingroup$ I used this book to teach a course this semester, the students liked it and it is a very good book indeed. [The book includes number of quasiindependent topics; each introduce a brach of graph theory and avoids tecchnicalities. I would include in addition basic results in algebraic graph theory, say Kirchhoff's theorem, I would expand the chapter on Algorithms, but the book is VERY GOOD anyway.] $\endgroup$ – Anton Petrunin Dec 7 '14 at 4:03
graphtheoryalgorithmsbook A GNUFDL book on algorithmic graph theory by David Joyner, Minh Van Nguyen, and Nathann Cohen.
There are lots of terrific graph theory books now, most of which have been mentioned by the other posters so far. I would particularly agree with the recommendation of West; one of the most complete and wellwritten texts there are.
But to me, the most comprehensive and advanced text on graph theory is Graph Theory And Applications by Johnathan Gross and Jay Yellen. Crystal clear, great problems and contains probably the best chapter on topological graph theory there is in any source by 2 experts in the field. It's pricey, but well worth it.
And of course, anything by Bollobas is beautiful. The problem with Bollobas, though, is that it treats graph theory as pure mathematics while the books by Gross/Yellen and West have numerous applications. Like linear algebra, the applications of graph theory are nearly as important as its underlying theory.

7

1
I know only one book on graph theory, Wagner, Bodendieck "Graphentheorie". It contains detailed introductions of the basic concepts and theorems and independent chapters on interesting special topics, the 3dr vol. is independent and on games, many exercises.
Who dislikes that may comment, because I use it for tutoring, mathcirclelike groups etc.
I heard good things about Combinatorics and Graph Theory by Harris, Hirst and Mossinghoff (Springer Undergraduate Texts in Mathematics).
It's easier to read than Diestel or Bollobas, but not dumbed down. But of course, it's neither thorough nor exhaustive.
Hajnal Péter, ''Gráfelmélet'' (Polygon, Szeged, 1997) is a very good introduction to graph theory.

$\begingroup$ See also a more specific question with this answer: mathoverflow.net/a/95565/5340 $\endgroup$ – Zsbán Ambrus Jun 24 '15 at 11:20
Different peoples need different introduction. If anyone want a light weight smooth reading I'll recommend Wilson. Although it is easy read, it'll take you even to the matroids.
If anyone want a proof based intro, research paper like, then West is the best choice. Diestel, Harary are to some extent for intermediate/advanced readers.
Some encyclopedic resources are Handbook of Graph Theory, Handbook of Discrete and Combinatorial Mathematics, and are easy read.
By any means bondymurty ranks first, assumed that readers are not below average.