The Internet is the place where you can browse to different computers around the world. Servers can be owned by a real player, like you, or NPCs.
When logged out, the first thing you will see is the server's web content. It might display useful information if it's an NPC, or it can be tailored content if the VPC is running the Webserver software.
On the upper-right corner of the web content, you can see which type of server you are connected to. Check the list of NPC types.
When accessing banks, you might notice a different UI. You have the option to hack and login to the bank server, plus the option to hack and login a bank account.
When logged to a bank account, you can transfer it's money or change it's password (if you are the owner). Please note that transferring money will generate logs on each account's bank, i.e., if you are on bank A and will transfer to bank B, both banks will have your IP, the account numbers and the total amount transfered.
Important IPs will be listed on the right sidebar, as a bookmark tool.
List of important IPs: