Again, my two favorite characters: Draco the half-orc barbarian and Eodrid the elf wizard. Opposite ends of the spectrum.
Barbarian is awesome because its total power-fantasy. I get to Hulk-out and charge everything. And that's a good strategy most of the time in combat. Honestly, I've seen groups spend huge amounts of time over-thinking strategy before a combat and it all goes to hell anyway. Just get in and start fighting.
Wizard is awesome because you get to solve problems no one else can and it gives you the most character customization. Back when all I had was the red box the only class that was different from PC to PC (outside of ability scores and even then not much) was the wizard because they got to select spells. I'm also a big fan of customized spells, so when I DM I let player's go wild with spell descriptions if it fits a theme. As a player, its fun to write your own spells if the DM allows it (Eodrid's Prismatic Arrow was my favorite).