Sony PSP Battery Pack
Your PSP comes with a PSP Battery Pack that was installed at the factory by Sony before it shipped to you. This battery pack was specifically designed to fit the PSP, and isn't a standard size. If something goes wrong with your battery, you may be able to have it replaced if your PSP is still under warranty. Otherwise, you can purchase a replacement battery pack for it from many retailers that sell video game equipment and accessories, or directly from Sony's website.
Sony PSP Extended Life Battery Kit
If you don't feel like your PSP's battery is lasting as long as you would like, you can purchase Sony's PSP Extended Life battery kit. It's rated at 2200 mAh rather than 1200 mAh -- Milliampere hours -- which means it lasts nearly twice as long as the original PSP battery pack. Just like a laptop, however, how long the battery lasts depends on how you're using your PSP and managing power. If you've got the screen on full brightness and are constantly watching movies or using the Wi-Fi connection, your battery life will be shorter than if you turn the brightness down and don't have the Wi-Fi turned on.
Third-Party Batteries
Other companies, such as Pelican, make batteries for the PSP -- some of which are even external battery packs, rather than internal. Still other companies have designed other power-delivery peripherals for your PSP -- such as Nyko's charger case, which protects your PSP and has a battery built-in, which complements the power in your PSP's onboard battery, thereby extending its life.
Considerations
Any battery option you choose for your PSP must actually say that it's intended for use with the PSP. Otherwise, it's likely that it won't work with your device, because the PSP doesn't use a standard battery size, such as AA or AAA. Sony's battery packs were designed by the same company that designed the PSP, but can also be more expensive than other battery packs. No matter what battery option you choose, research it thoroughly before you buy. Also check whether there's a warranty offered for your battery, in case it doesn't work as expected.