Books about braid theory I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. 
Thanks for any suggestions.
 A: The only general purpose textbook that I know of is:

Kassel, Turaev, Braid Groups, Graduate Texts in Mathematics 247,
  Springer.

I think it's the standard reference today. It's good, but it's not the funniest book in the world.
A nice collection of survey articles is

Berrick, Cohen, Hanbury, Wong, Wu, Braids: Introductory Lectures on Braids, Configurations and Their Applications, World Scientific.

I haven't read all of these articles in detail, but Rolfsen's "Tutorial" is nice and I liked Cohen's and Ghrist's articles a lot (but YMMV).
After that, it really depends on what you're interested in. Braid groups are really versatile, and different people study them with different tools and from different cultures (that's probally the most exciting thing about them). A quite easy book on the links with Knot Theory is 

Prasolov, Sossinsky - Knots, Links, Braids and 3-Manifolds, Translations of Mathematical Monographs 154, American Mathematical Society.

but, really, any textbook on Knot Theory worth its price will talk about braids at some point.
If you are interested in more combinatorial/algebraic things around the left-ordering of the braid groups, I strongly recommend

Dehornoy, with Dynnikov, Rolfsen, Wiest, Ordering Braids, Mathematical Surveys and Monographs 148, American Mathematical Society.

Even if the left order on the braid groups isn't what excites me the most about them, I remember enjoying the beginning of this book quite a lot. Dehornoy is really a superb writer. He also wrote a number of survey articles (available on his webpage) and one of them,

Dehornoy, Notes on the Braid Isotopy Problem, lecture notes.

is also a great place to learn things about braids.
If you understand French, there are also a bunch of videos on the same theme on this webpage.
The most accessible references in this list are probably the Prasolov-Sossinsky book and the Dehornoy notes. They are interested in quite different aspects of the theory so you could (should?) try to read both. 
The other references are probably a bit too advanced to be read by an undergraduate from cover to cover, but I'm sure you could get some perspective by trying anyway.
A: Let me also recommend the classic "Braids, Links, and Mapping Class Groups" by Birman. It might end up being a little more advanced, but it's worth a look.
