How can I understand which arduino I need to run some code?
In this case, the description on Github says specifically that it is compatible with the Uno: 


  Based on the work in
  http://www.instructables.com/id/Use-an-Arduino-with-an-N64-controller/
  here comes a comfortable library for usage with e.g. Arduino Uno.


If you know in advance that you intend to write a very large body of code, of which this library is only a part, you can anticipate needing an Arduino with more memory at some point, but an Uno will get you a long way there. And, with all due respect, the fact that you're asking this question also suggests that an Uno will be more than adequate. 

Genuine Italian Unos just barely exceed U$D 20 and Uno clones are about 2/3 of that price. Some of the clones may not be quite up to quality of the genuine boards, but reputable board makers' products are usually quite good. The best advice I can offer would be to buy a genuine Uno for your first board, both for the most likely best experience starting out, as well as to support the folks who did the upfront work and released the design as open source. Then maybe branch out to other board-makers' products for price, or especially, for added features (assuming those are useful to you) that some of them have engineered into their versions of the product.

Update:


  My final goal is to put it into a Nintendo 64 so its size is really
  important. Something like this could be ok?
  kiwi-electronics.nl/beetle-arduino-leonardo


The Beetle's processor, AtMega32U4, has a similar memory complement as the Uno's AtMega328. In fact it has a little bit more RAM than the Uno. It should be quite doable to compile it for the Beetle.