iPods, Zunes, iRivers, Onyx, etc. are all MP3 players - they're just built a bit differently, have different displays, and use different programs to put songs on them.
The only real difference you should pay attention to when buying an MP3 player is how it works - is it a spinning hard drive, or is it a non-moving hard drive inside? When information is written to iPods, it can either be written to a spinning disk (much like how information is written to your computer's hard drive) or it can be stored in a solid state (much like how information is stored onto thumb drives, if you've ever come across one).
If you buy an MP3 player with a spinning disk inside, it's not meant to be wiggled around or thrown anywhere - that makes the disk skip, and could damage your MP3 player. If you want to jog with music, I would recommend getting a solid-state player so you don't run that risk.
Another confusing thing is gigabytes (GB). A song tends to take up about 5MB (megabytes) of space nowadays, maybe even up to 7MB. There are 1,000 MB in a GB. So if you buy a 4GB iPod, you're buying space for about 800 songs. That should give you an idea of how much those 160GB iPods are holding!
As for the difference between all the MP3 players, right now the iPod, the Zune and the iRiver are holding the market as most reliable. Any of those three will certainly last you for years. But feel free to check out any of the other MP3 players, and ask the "little store helpers" what they think.
Best of luck on getting your MP3 player! :)