There are free apps and apps that you pay for. These apps come from various sources. There are individual developers and commercial developers, so do be careful when using an app from an unknown source as they can contain viruses.
One of the best PSP apps is PSP Video 9; it is a video conversion utility, giving you the ability to convert various video formats into a new and different format. This comes in handy if your video player only plays certain formats, or you have downloaded from the internet. This is a free app so you can try it without worrying.
BookR is an awesome free PSP app that allows you to read all of the major text formats on the PSP. These include PDF, text files and all the rest. If you use your PSP much at all outside of the gaming realm, this is a must have because you will surely run into text files along the way.
There is the PSP PDA, this is truly awesome, it gives you the ability to use your portable game system to track appointments, keep addresses and emails, phone numbers and so much more. This is also a free app available for the PSP giving you a true portable and useful app experience.
PSP radio player gives the PSP the ability to access many of your favorite stations without the need of the radio adapter. This is free and is definitely in my top ten. I can basically have radio anywhere I want with just my PSP and nothing else. It is one of the best PSP apps out there.
You must have the PSP Media Player. PSP in itself can only play video that is in the PSP format, but with Media Player you can watch videos in all the common formats. Combining this app and the video conversion utility gives you the ability to watch almost any video you can get your hands on.
PSP VNC is a great program for multi-computer homes. If you have wireless in the household VNC lets you access the other computers in the home from the PSP and vice versa. You can click over and check your email, see how a download is going, maybe even sneak in and see what the kids are playing on the PSP.
This is only a small portion of the apps available for the PSP. Accessing homebrew applications also opens up many, many more options, but keep in mind that most homebrew apps are not the prettiest in the whole world, but they do some awe inspiring things.