We don't have any trams here (though there are always talks of bringing them in) so it's pretty much all buses to get around town! The good news is, we have several great bus companies working at reasonable prices, so you should be able to get where you need to go, whenever you need to go.
If you're here for a while, it's worth getting The Key - a smartcard for travelling on the buses that you can top-up online or in the One Stop store in Brighton. It offers cheaper rates for everyone and even more so for registered students - see here.
Most bus services are run by Brighton & Hove Bus Company - they're the ones in cream and red. They are known for being particularly environmentally friendly, now running all their buses on biodiesel and they have won quite a lot of awards UK wide over the past few years.
Stagecoach, Metrobus, Compass and SussexBus all run some services in Brighton too, though they tend to run further afield, or to other cities and towns.
The Big Lemon bus company deserves it's own mention too though. They currently run 3 routes, one from the uni campuses in Falmer, one between City College campuses and one between Brighton and Woodingdean. They are a social enterprise, running their buses entirely on waste cooking oil (hoping to go all electric in the coming years) and focusing on happy drivers and pleased customers. You'll always hear a good morning or good afternoon when you get on and they'll often wait if they see you running, making for a nice atmosphere!
In terms of timetable and frequency, buses in Brighton are generally very good. Most places have buses into the centre at least every 20 minutes and you can go from there to pretty much anywhere in the Brighton area or beyond. Night buses also run on most main routes for almost the entire night (last buses are usually at 4 or 5am and they start again at 6 or 7). Prices are OK - they're not the cheapest buses, but they're not bad. A single journey costs £2-2.50 and a day ticket costs £4-5. If there are several people travelling, we'd suggest you pool together for a cab, since you can save money and time, particularly if you fill the cab up. You can find cab info here!