A.1. Legacy System Functions (a.k.a. Global Variables)
Returns the number of connections attempted since the last time your SQL Server was started.
This one is the total of all connection attempts made since the last time your SQL Server was started. The key thing to remember here is that we are talking about attempts, not actual connections, and that we are talking about connections as opposed to users.
Every attempt made to create a connection increments this counter regardless of whether that connection was successful or not. The only catch with this is that the connection attempt has to have made it as far as the server. If the connection failed because of NetLib differences or some other network issue, then your SQL Server wouldn't even know that it needed to increase the count — it only counts if the server saw the connection attempt. Whether the attempt succeeded or failed does not matter.
It's also important to understand that we're talking about connections instead of login attempts. Depending on your application, you may create several connections to your server, but you'll probably only ask the user for information once. Indeed, even Query Analyzer does this. When you click for a new window, it automatically creates another connection based on the same login information.
This, like a number of other system functions, is better served by a system stored procedure, sp_monitor. This procedure, in one command, produces the information from the number ...