Both JR and Keisei operate "luxury" trains (Narita Express and Skyliner/Cityliner, respectively), and "ordinary" trains, to the centre of Tokyo. The most expensive is the JR Narita Express (about 3000 yen), but it is has the advantage that it goes all the way to Shinjuku and to other destinations in the Tokyo area. The Keisei Skyliner costs 2400 yen and is faster, so it is a good alternative if you don't mind the extra journey from Nippori to Shinjuku. To buy a ticket you have to ask specifically for a Narita Express or Skyliner ticket, and you will get a reserved seat. There is also a slower Keisei Cityliner which costs 1920 yen to Nippori or Ueno.
For ordinary trains (i.e. not Narita Express or Skyliner/Cityliner), there is only one kind of ticket, regardless of the classification of the train (local, express, etc). You can buy these tickets at the ticket counter or from machines. The cost to the centre of Tokyo is in the range 800-1200 yen.
The JR ordinary train is the next most expensive. It goes to Tokyo station (which is less easy to negotiate than Nippori), and it is relatively slow (80-90 minutes from Narita Airport to Tokyo). The Keisei ordinary train is probably the best "economy" option, (around 800 yen, 70 minutes).