I've got an olympus Camedia C60 that runs on a rechargeable battery. It's a discontinued model now so that's not too much good for you as you'll more than likely find out for most of the specific models for the less expensive P&S cameras. I got mine for about $150 and it's a 5MP camera if memory serves me right. Uses a rechargeable Lithium battery that's good for a weekend of 500+ pictures. Has optical & Digital zoom with the ability to block use of digital zoom.
I'll presume that you want a P&S camera as opposed to DSLR. Basic facts to compare are.
1. Optical Zoom, (not total zoom just the optical portion, If it doesn't say optical it's not)
2. MP capability. If you're looking to print standard a 4X6 then 2-3MP (I have an old 1.2MP that works acceptably, upon close examination I can see the difference but as a glance at the picture you don't notice it) is great when you start going up in size start going up rapidly and minimize or eliminate use of digital zooms. Pay attention to a digital zoom as it zooms it decreases the effective MP of the camera (2X digital zoom just took your 4mp camera down to a 2mp effective)
3. Batteries - matter of personal preference here, standard AA batteries do not last long even if you buy the expensive super dooper made for electronics batteries.
If battery life is important to you then find a camera that has a traditional viewfinder. Using the LCD screen on the cameras chews up battries very very quickly.