Your Playstation Network ID is your handle when navigating PSN content and online gaming on the PlayStation 3, an identifier as unique as you choose. As long as your "unique" handle is actually available, that is. Since PSN has a huge number of players, any name may potentially already be taken. There are two ways to check for availability: Try to sign up using the name, or do a search of PSN public trophies.
The simplest way to check the availability of your desired PSN ID to attempt to sign up using it. If the name is already taken a message will pop up to that affect and you will be prompted to enter another name. You can come up with a different name, or create a variation on your original name by adding symbols, changing the spelling or swapping out letters with numbers.
PSN Public Trophy Search
You can also check ahead of time to see if your preferred PSN ID is available by going to the PS3 website and looking up the name in the Public Trophy search. If the name is taken, a list of public trophies earned in their games will be displayed. If not, it'll tell you the user you're looking for doesn't have a profile.