OK, 'tis the season...
What is AppServer?
A headless ABL client. It can run procedures for other ABL clients and it can connect to databases for CRUD operations. That's a very incomplete definition. For a good one, read the Progress documentation. There are several manuals that cover AppServer.
What is WebSpeed?
Conceptually it is very similar to AppServer, but it is focused on building web applications. Again, there are several manuals that provide all the information you could want.
Why do you need to stop brokers to back up the database?
You don't. You didn't specify why you think you need to do this or how you are backing up the database. If you are using the Progress backup utility (probkup) to run an online backup, it can safely back up the database while clients (including AppServer and WebSpeed agents) are connected. If you are using some other method of backup, like a SAN snapshot, you should first quiet the database (using proquiet) but you still don't have to terminate agents or brokers. For more information on backup up databases, read the Database Administration manual.