LAN = Local Area Network
Basically a bunch of connected computers/devices for the purpose of sharing data, physical resources (stuff like printers) or even processing power in some cases. The main use of a LAN network for the standard user is to supply an Internet connection to devices. Though there are other things like a NAS (Network Attached Storage) which can be used to backup local machines or just generally hold a ton of data which can be accessed by devices on the network.
The Internet would be classified as a WAN (Wide Area Network), which typically covers a far larger geographical area than a LAN, though is technically defined as two or more LAN's connected by a 3rd party connection/s.