There are a bunch of different kinds of batteries, and trying to pick one for a DIY project gets confusing fast. DIY enthusiast afrotechmods made a video explaining the basics of different batteries, as well as a guide to picking the right one for any DIY project.
The video goes through five different choices of batteries and talks about the pros and cons of each. For example, lead acid batteries are cheap and durable, but very heavy. Lithium polymer batteries are light, but they're expensive, require extra management and can be explosive if handled improperly. Even if you're not a DIY type of person, the video's worth a watch just to get the basics of battery chemistry into your brain.