Bettas are one of the few fish that can be kept without a filter; but we don't recommend it. All fish do better, will be healthier and live longer in a properly filtered and cycled tank (yes, we're going to have to explain 'cycling' now, I know; 'cycling' is growing a colony of the good bacteria, that akasha mentioned earlier, inside the filter. Those bacteria take care of the fish's waste's for you, so you don't have to be doing water changes every day and, most importantly the fish is protected from a build up of ammonia that can kill it).
We never recommend tanks of less than 5 gallons,