Homebrew technically refers to original or individually ported games that have been written/produced by people not liscenced by a console's maker. This could be anything from a clone of pong to a port of quake, or even an ebook reader program.
These days emulators tend to be referred to as homebrew also as they are probably the most popular form of unofficial software that people wish to run on a console. When I use the term I mean any unlicensed program that you may wish to run on your console.
You have a firmware 2.71 psp, which is great, but you also need to check one other thing before you start thinking about which method to use to allow you to run both homebrew and any commercial UMD games you wish to buy.
You need to check to see if you have the newer hardware version, known as ta-82. You can check this looking inside your psp's umd drive door as shown in
this image.
If your psp is NOT a ta-82 you have 3 options:
1) Downgrade your psp to firmware 1.5 (this is method I currently use). This allows you to run homebrew/emulators straight from the psp's 'game' menu, however you only have the features of firmware 1.5 (no web browser) and cannot play recent game UMD's.
This would be terrible if it were not for a brilliant program known as devhook. This program allows you to load and run any of the more recent firmware versions, giving you access to all their features and the ability to play every UMD game released.
It does this by loading the newer firmware into memory from your memory stick, instead of installing it into the psp's internal memory. Once you turn the psp off and back on, you'll be back to firmware 1.5.
This means you can keep your psp at firmware 1.5 and have the ability to run any homebrew you wish, but then load easily into a more recent firmware to play recent games/use more recent firmware's features like the web browser, rss feeds and whatever else.
2)Install a custom firmware 2.71SE-C. This is a modified (unofficially) version of firmware 2.71 which basically means you get all the features of firmware 2.71 and its ability to directly load all (except for the most recent of them) game UMDs, and also keep the homebrew access of firmware 1.5. To play the most recent of game UMDs you may need to use devhook, but this is very easy to do.
3)Keep the standard firmware 2.71, but install a small patch (data modification) known as HEN-D to the psp's internal memory that stores the firmware, which allows you to run some (not all) homebrew. One of the things it does allow you to run however is devhook, through which you can load (temporarily, untill you turn off the psp) firmware 1.5, and have access to homebrew that way. Kind of a reverse-devhook, as its normally used to load higher firmwares not lower.
Personally I would recommend option 1, many people love the custom firmware however so I guess you should look into both if your psp is NOT a ta-82.
If your psp IS a ta-82, your only option currently is number 3, as this version of psp hardware has been designed to prevent downgrading to firmware 1.5.
The best place I know of to look for information is probably
this site. You can see a list of all the emulators available on the menu bar down the left hand side, or use the psp help forum there to ask questions.
Feel free to post here or PM me if you have any questions.