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Cyan for Wind Godness
Sabin for OMGZ 9999 DMG VS ANYTHINGSness
Strago for "Hey guess what, I just Hasted, Shelled and Protected the whole party before you could even do anything"ness
and Gau for the "Are you trying to kill me? I can't feel you behind my Snow Muffler"ness.
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Relm with Soul of Thamasa and Celestriad. Highest magic in the game lolwutdead.
Sabin because he's totally awesome and sexy.
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Usually have Celes and Edgar in the party because their abilities in battle are really quite handy at times. I've always quite liked using Cyan and Sabin too, but they're not definites because I really like to chop and change in FFVI.
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Terra is probably the best character in the game. She has the 2nd strongest Magic stat in the game and has good Attack and Defense as well. Not to mention Morph, which doubles her Magic and Physical damage and reduces the damage she receives in half.
Locke is the fastest character in the game. I, being a fan of Speed over all other stats, think he's awesome.
Edgar is a great character in the beginning for his Tools, which seems to be the main reason why people like him. But that's not the only reason why he's good though. Drill and Chainsaw are all good in the beginning, but once you get near the end of the game, there's much better ways to deal damage. The best is probably Dragon Boots + Dragon Horn.
Sabin. Pfft. The only thing he has going for him is Bum Rush.
Celes' stats are very similar to Terra, except she can deal slightly stronger physical damage than Terra while she does slightly less magical damage. In my opinion, she's a great character who is really worth using. The only problem is that some might just choose Terra over her.
Shadow can use his Throw can usually let him do high damage. Also has the nice touch of Interceptor counter attacking. An fairly good character, maybe even better if you could use him more in the WoB.
Cyan is so overrated, about as much as Sabin. His best attacks require charge time that's not even worth the wait. Quadra Slam is the only thing that's somewhat worth the wait, though by then, you probably could have already dealt more damage with your other characters. Dispatch is the only thing actually worth using, which is slightly stronger than his physical attack.
Setzer isn't one of the best characters, but he isn't as bad as, say Sabin or Cyan.
Strago has really cool Blue Magic, which can be very useful. He'd be a really great character if his Magic stat was a little higher, like Terra's Celes' or Relm's.
Relm's Sketch sucks. But since she has the best Magic stat in the game, that makes her very nice to have.
Mog is an average character. His stats aren't that special, but he can use Dragon Boots + Dragon Horn. Moogle Charm is also very nice.
Umaro is the worst character in FF6. Oh yeah, he's got good stats and useful in the Fanatics Tower, but nothing more. Other characters have more variety and can use things such as Magic and Espers and aren't in some kind of Auto-Beserk status.
Gogo is the most useful character because of his amazing versatility and Mimic. Only problem is that his stats are quite low.
I don't even know what to say about Gau. I guess if he was one of the monsters in the dinosaur head-shaped forest he could be really kickass.