When people first set up a home network they sometimes wonder what all the IP addresses they see are. I will try to explain some of the more common ones you are likely to see.

Public IP address

Also known as your WAN IP or external IP address. Every computer or machine on the Internet must have a public IP address and its how other machines will recognise and be able to respond back to your PC. Go here to lookup your Public IP address.
Your public IP address will normally be assigned to you by your ISP and can either be static (stays the same) or dynamic (different each time you connect). - See Static IP -v- Dynamic IP

Private IP address

Also known as your LAN IP and is used by machines on a private network. Each machine on a Local Area Network will have its own private address which is used by other machines on that same network to communicate with each other.

From ipconfig /all

Physical Address. . . . . . . . . : 00-0C-76-A8-B4-46 Dhcp Enabled. . . . . . . . . . . : No IP Address. . . . . . . . . . . . : 192.168.1.2 Subnet Mask . . . . . . . . . . . : 255.255.255.0 Default Gateway . . . . . . . . . : 192.168.1.1 DNS Servers . . . . . . . . . . . : 212.159.13.50                                     208.67.222.222

Physical Address

Also known as your MAC (Media Access Control) Address. Every network device such as a Network card or Wireless Access card will have a special address which uniquely identifies that particular piece of hardware. The MAC Address is given to the device by the manufacturer of the hardware. If a PC has 2 network adapters such as ethernet and wireless then each will have a separate address.

DHCP Enabled

(Dynamic Host Configuration Protocol) The example above shows dhcp disabled, but many home routers will also act as a DHCP server and automatically dish out IP addresses to any PC or device that tries to connect to it. If DHCP is enabled then the router is responsible for "leasing" LAN IP addresses to Computers on the internal network. ipconfig /all will also list the IP address of the device which is responsible for giving the PC its LAN IP - eg

DHCP Server . . . . . . . . . . . : 192.168.1.1

IP Address (LAN)

The machines LAN IP address (see above).
IANA reserves special IP ranges that can be used for LAN IP addresses

10.0.0.0 - 10.255.255.255
172.16.0.0 - 172.31.255.255
192.168.0.0 - 192.168.255.255

Subnet Mask

Each LAN will have its own IP range. In our example it is 192.168.1.0 or 192.168.1.x
By using a subnet mask of 255.255.255.0 this allows us to divide our network into 255 IP addresses in the range 192.168.1.1 to 192.168.1.255
In reality its a bit more complicated, but I'm trying to break it down into the most simplest of terms to aid understanding.. and doesnt take into account such things as 255 is a special broadcast address. More info

Default Gateway

The next hop on our journey out towards the Internet, therefore for a PC on a LAN, the default gateway is most likely to be the IP address of your router.

DNS Servers

Domain Name System Servers are responsible for resolving IP addresses into a human readable format.
Its the job of DNS Servers to translate IP addresses into domain names such as 64.233.183.104 into google.co.uk. Without DNS our web-browsers such as Internet Explorer wouldn't be able to navigate to the correct website when you type in www.domain.com. ISPs usually have at least 2 DNS servers which will be assigned to you, although you are free to set any DNS servers you like in your network config. In the example above I have assigned which DNS servers the PC uses.
If DNS hasn't been set on a PC, then the local machine will pick up any DNS Servers automatically assigned to the router by your ISP. In cases such as this then ipconfig will show the router as being the DNS Server.
             DNS Servers . . . . . . . . . . . : 192.168.1.1

WAN IP Address

Public IP address visible to the Internet - See above for more info.

Network Address Translation (NAT)

Using Network Address Translation (NAT) our router is very cleverly able to identify which PC requested data and forward it on to the correct LAN IP address. To the outside world our router is the device with the WAN IP address and if set up correctly, a NAT router wont forward any unsolicited data to any of the machines on the private side of the network.

Internet <-----> WAN IP ||Router|| LAN IP <-----> PC

LAN Address

The routers LAN IP address as it is visible to any machines on the private network.

Default Gateway

The next hop for traffic out on to the Internet - in this particular case the gateway on your router will be your ISPs edge router.

DNS Servers

See also above for explanation of DNS Servers.. but here we can see they are different from the ones assigned on my local machine. This is because the router has picked up the DNS routers assigned by my ISP. The DNS servers set on the local machine will over-ride the servers automatically assigned by the ISP. If DNS hasn't been specified on the local machine, then the PC will use the ones assigned to the router.