Elf are PS2 executable files, just like exe are executable files for PC comp's, Elf is the executable files for PS2.
IRX files are what some PS2 progs need to be able to run. IRX files are similar to DLL files.
PS2MENU is a homebrew loader/copyer. This prog allows you to run/copy homebrew software from Mem card, HDD, CD, et al.
PS2LINK is a loader program which loads progs thru the network adapter. You run PS2LINK on your PS2 & you run a client prog on your PC. Your PS2 & PC must be connected thru Ethernet for this to work.
I would suggest doing a Google search for info the progs you are asking because if I or anyone else tries to explain everything you want to know about homebrew, it would take too long to mention.
Homebrew Supporter for over 4+ years.
PS3/XBOX 360 fan (own both systems)