Nowadays majority of trail cameras are powered by either C-cell batteries or AA batteries. The benefit of C-cell batteries is that they have higher capacity than AA batteries. Thus when operating in the field temperatures over 32°F they can last months. The downside is that cold weather affect to C-cell batteries; in freezing temperatures battery life is easily cut in half.
The biggest benefit of AA batteries is that you can use rechargeable Lithium or Nimh batteries. This will save you tons of money in the long run and what’s important, Lithium and Nimh batteries are generally unaffected by cold weather.
In addition to battery type and promised battery life, you should also check the possibility to use external power sources to power the trail camera. Many trail camera models can be powered by external battery packs or solar panels.