Peltiers aren't very efficient; a phase-change heat pump system (like most household refrigerator/freezers and air conditioning systems) is 6-12 times as efficient
, and on their own TECs can't move heat nearly as far as a phase-change system can. They're really only good for very localized cooling (overclockers used to (still?) use them on CPUs to squeeze another hundred MHz or so out of their machines), or situations where you want heating and/or cooling (especially both) in a small, lightweight package.
If you want to get a sense of how much cooling power you have:
Total power * efficiency * time = energy
50W * 0.10 * 1hr = 5Whr
5Whr = 4.3 kilogram calories, so your 50W TEC refrigerator could cool a kilogram of water by up to 4.3°C in one hour, not accounting for losses due to imperfect insulation. Of course that also assumes your specific TEC is 10% efficient, which it may not be.
One big advantage to peltier devices is that you change direction of heat flow with the direction of current flow. There are commercially produced travel coolers that can run from a car or wall adapter with unpolarized DC plugs--mate the plugs one way and the container keeps your drinks cool, or mate them the other way and it'll keep your casserole warm on the way to grandma's house. So if you really wanted to develop an arduino project, you could do something similar with both warm and cool temperature control.