What (good) is MD5?
If you download a file, you have a couple of things to worry about. Did the file get transferred and saved without errors? Is the file you received really the file you needed or some modified bogus copy? Did you get the proper version of the file?
MD5 helps you verify all the above. With EVE Launcher there's additional concerns. Can this Grismar character be trusted? Why is this piece of software registering my keystrokes and what is it doing on the network? MD5 doesn't help with that, whether you trust the software once you're sure it's the genuine article, is up to you.
MD5 is an algorithm that generates a unique hash (also called checksum, or md5 code) for a given file. Any MD5 program will generate that same hash for that specific file. An MD5 hash looks like this: 131A5541B31D676FC4775D08323AD3C5. Note that the capitalization of the letters (A-F) in an MD5 hash does not matter. So, 131A5541B31D676FC4775D08323AD3C5 = 131a5541b31d676fc4775d08323ad3c5.
Now, if someone you trust tells you the MD5 hash for a file, you can take that file and some program to generate an MD5 hash and compare the hashes. If they match, you can be sure you have the proper file. If they don't, something went wrong (network, harddisk failure) or someone messed with the file (virus, trojan, error during previous handling).
Examples of programs for generating MD5 hashes are Ferruh Mavituna's MD5 Checksum Tool or Mark Gillespie's MD5 Checksum Tool (recommended). But I would encourage you to find one of your own and not take my word for it.