My experience is with Carnival Cruise lines. It may be different on other lines.
The two biggest things that will run up your tab are drinks and shore excursions.
Drinks: Expect to pay $2-$4 for a beer or coke, and $7-$12 for a mixed drink. You can often buy a "Fountain card" or something like that that is a fixed price for sodas throughout the cruise. Probably a good deal if you are a big soda drinker or are with kids. You get juice, iced tea and water at no charge. Oh yeah, and they add an automatic tip on every drink. I'm sure you could ask them to take the tip off, but you'd look like a tool for doing it considering the (mostly) Phillipinos who serve you drinks are probably working 16 hour days for peanuts.
Shore Excursions: These can really add up. You could easily double the price of your cruise if you do a lot of the higher end ones. You can usually save a ton of money by not booking them through the cruise ship and just pre-arranging them through the internet or booking them at the kiosks that are inevitably right next to the dock at each port. The cruise director will tell you all sorts of scary stories to dissuade you from doing this, but it is mostly bunk. Often the cruise ship goes through the exact same tour companies and they add as much as a 2-3x markup.
The Fancy Dining option: There is usually a high-end restaurant that you need reservations for and charges a $25-$30/person. I never bother with them. The regular dining options are good enough for me. But if you are a surf & turf aficionado you might want to budget for that.
Internet is super expensive, around 70 cents a minute if I recall. You can buy bulk minutes, but it isn't that much cheaper. You are better off just bringing a laptop or smartphone and using them in ports. I had no trouble finding free wifi at most ports.
Cell Phone: Danger Will Robinson! Your cell phone will work on-board, but be prepared for a huge bill when you get back. I STRONGLY suggest turning off your phone on board or at least disabling the ability to receive calls and text messages. You can passively run up a bill that way.
Tips: This is one thing I like about cruises. They just add about $10/day/person to your bill for tips and there is an option at the end of the cruise to add to that tip for the room steward, maitre'd, etc. There is absolutely no pressure at all to add to that tip, i.e. there is no one standing with their palm out.
I never minded tipping for service, I always just hated having to carry cash around to do it. They make this process very easy on board. You really don't need to have cash on board at all.
Bottom Line: I tend to buy at least a few drinks/day and find my own shore excursions. With a few souvenirs thrown in here and there I typically spend about 50% of the cruise price in extras per person.