Is a private key tied to a user name or is it the user name?
No. There are no user names necessary.
And you definitely don't want to treat the private key as a user name because you want to keep that private. It is what allows you to control the bitcoins.
Private keys can be generated instantly, and in great number. For instance, if you wanted a million private keys, you could have them. When people talk about a bitcoin "wallet" they are essentially talking about the private key.
Now, there are corresponding public keys mathematically associated with each private key. And you can generate a large number of public keys for each private key (I think there is a limit, but I don't remember what it is, but it's a pretty large amount).
So, for instance, you could have "wallets" like this (note: these are illustrations, this is not what the keys actually look like):
Private key: 12erhjfh30847y50234jrqhewfi23g49fgerugf29374t982734gfqhf982348rtg234r
Corresponding public keys:
If you send bitcoins to any of those public keys, they will arrive in the wallet of that private key.
And so on and so on...
So you can generate as many wallets as you want....
and within those wallets, you can generate numerous public keys
All with NO registration
NO user accounts
This post was edited on 11/30 at 2:29 pm