The "Auth to Services" option lets you log in when you have registered with Freenode's NickServ, which reserves your nickname and provides other privileges.
This SuperUser post has a guide for signing up with NickServ with webchat if you haven't done so before.
You haven't said which IRC client you're using, so I'll be using Freenode's Kiwi chat as an example.
Most IRC clients will have tabs for each channel you're in and each private messaging conversation you have with a user. They will also generally have a Status channel. This channel gives information about the server and your connection to said server.
Kiwi IRC dev here—the network tabs not closing when you issue a /quit is a known bug right now and should be fixed soon.
It will most likely in the next release in the upcoming week, which includes closing networks via the UI also.
Your IRC traffic is never logged. Not even to kiwiirc.com servers. This ensures that no matter who asks or who looks, the IRC traffic can not be read elsewhere.
The only logging information I can see in the source code is error logs.