It depends on what kind of battlebot you're making. I'm not an expert on Battlebots, but I assume there are two standard approaches: surviving anything, or avoiding everything. Thick, heavy armor will make your battlebot slower and less maneuverable, but strong, heavy bots are better at pushing their opponents around and strong armor will withstand damage better. Thin, light armor will let your bot zip around quicker and run circles around a slow opponent, but don't provide the solid foundation for a heavy, powerful weapon.
Once you've decided how heavy your armor can be, the other, and often largest, concideration in choosing armor material is price. Steel is strong, cheap, and heavy; aluminum is stronger and lighter for the same volume, but might cost more. The main section on this site has info on common materials here
, including detailed articles on each. The only other thing I remember seeing used was plexiglass (or something like it); I imagine it would be most expensive of all, but cool-looking and probably very light-weight.
Some advice: The simplest, and often most effective, designs are wedges; these often strike a balance between speed and armor, and their strategy is basically sumo; the low front gets under enemies and tries to lift them off the ground, allowing them to be pushed around into stationary hazards. This can be effective against almost any kind of bot, and their simplicity make them a great place to get started.
Good luck with your project!