Understand the OSI 7 layer model of networking, once you understand that, the rest of the pieces fall together easily.
The Internet is THE public IP network. There is only 1.
The Internet is the interconnection of Intranets.
An Intranet is a private IP network. There can be 1000s
An Extranet is accessing an Intranet over the public IP network (Internet).
VPN the technology used on the Extranet to secure it.
LAN = Local Area Network. Despite the answers above, a LAN is in reference to the physical network. A LAN is limited to a single facility, such as your home or school or building you work in. A LAN can use any layer 2 technology (Ethernet, Token Ring, etc..). A router separates broadcast domains on a LAN.
Broadcast domain = the part of the network where if a device sends out a broadcast frame, it will be seen by all devices on the LAN. Routers separate broadcast domains. An ARP request is example of a broadcast.
WAN = Wide Area Network = a network that usually connects multiple LANs together. Historically a WAN used some other layer 2 protocol than Ethernet, such as Frame Relay. A WAN is L2 / L3 protocol agnostic, the term is intended to mean your are connecting two or more LANs that are separate by a long distance - like NYC to LA.
Intranet is privately owned network that uses the same protocols and tools as the Internet. It's a private organizations IP network. A big company would have it's own Intranet, where the employees access internal websites via tools like Firefox and it uses all the same security mechanisms that the Internet uses. But it's only accessible to the companies employees. An Intranet can be connected to the Internet, but the Intranet part is firewalled off from the Internet.
Extranet and VPN (Virtual Private Network) are sort of one in the same. A VPN implies a secure/encrypted connection from one private network to another private network over a public network. A VPN can be a piece of client software on a PC, connecting securely to a VPN gateway in order for one user to gain access to a company Intranet. A VPN can also be a router running on LAN, that creates a secure connection to a VPN gateway to allow all the users on the LAN side of the router to connect to the company Intranet.
If it helps, in the real world we don't use the terms Extranet or Intranet anymore. We use public and private network.