Final Fantasy XIII Platinum Trophy. Max out all characters. The. HORROR.
Sorry. Doesn't help much if you've never played the game. Others have answered your question, but here's my two cents. Grinding means when you do certain tasks over and over and over and over and over. Normally, these tasks and small and annoying errands. Go kill a level 1 monster again and again, even when you're at level twenty. People often do this when they want experience points (in which case the monster would be a higher level than 1), but there are other reasons to grind. One such reason is farming, in which you kill a certain creature, or play a certain mini game, or whatever, again and again in hope that you kill the ultra-rare item that can only be obtained by doing said action. Sadly, the odds of getting the item are normally 1/100,000.
There's my two cents. No idea if it helped, but grinding is rarely fun, and always annoying after ten minutes of it.