ATAF is the local bus company in Florence - their ubiquitous orange buses (and some are now red too) are everywhere - some are big, some are small, some run on natural gas, and some run on electricity. From their website:
"ATAF, Azienda Trasporti Area Fiorentina, represents a consortium of nine municipalities: Bagno a Ripoli, Calenzano, Campi Bisenzio, Fiesole, Florence, Impruneta, Scandicci, Sesto Fiorentino, and Vaglia, that cover a total territorial surface area of 538 square km with a population of about 600,000 residents."
The site has a ton of information including route maps and timetables, and news about strikes, etc. The fares recently went up - almost all guide books and most websites have the old fares. The current basic fare is 1.20 €, for a 90 minute ticket (you buy the tickets at tabacchi stands or ATAF kiosks) - you must stamp the ticket when boarding (even if no one else seems to - you don't want to get busted without a validated ticket - the fine is steep!) and it is good for 70 minutes - you can make as many transfers as you want in the allotted time. A ticket with 4 rides is 4.50 €, which saves you .30 €. You can also now buy tickets directly on the bus, for 2.00 €, at night only, I think - this may have changed recently too as they try to become more customer "friendly".
There is no central bus station per se, the routes all have their own beginning and ending point (capolinea), but many buses have Santa Maria Novella train station as one terminus. The A, B, C, and D are electric mini buses that run routes in the Centro Storico of old Florence and the Oltrano. All other buses have numbers. A couple to know for tourists are the 7 to Fiesole (the cheapest way to get up there) and the 10 to Settignano, both of which terminate at the train station. There is a very good route map available, but it can be hard to find one. Try the main ATAF booth at the train station, or the tourist office near the Church of Santa Maria Novella.
Update: If you have an Italian cell phone you can also send a text to get a ticket - pretty handy. Text "ATAF" to 488 0105 and you should get a confirmation back in moments.