What is a router some of
you may be asking?  Well the simplest way to explain it is it is a computer
with two network cards in it.  This computer, the router, will route a request from one IP subnet to the another.  A subnet is a group
of IP addresses that share the same properties.  Each IP address is made up
of 4 octets.  An IP address is made up of two groups: a network ID and
a node ID.  When an IP address is given to a computer, it must also
be given an IP subnet mask.  This subnet mask tells the computer if the
octet belongs to the network or node ID. An example is 255.255.255.0 (Class
C).  The three 255 octets tell the computer the first 3 octets belong to
the network ID.  Also the computer can be given the address of a gateway. 
The gateway is a router.  It routes all information for a computer on a different
subnet to this gateway computer.  Below is a picture of how it works. 
Two computers on separate network IDs can not directly connect to each
other.  This is where the router comes in:  

What is NAT?
These routers are different than most routers. 
Since the LAN IP is a private IP, the router must use NAT to request information
of a website.  This translates your request to the public IP, which is the
the WAN, and then requests the information.  When the information comes
back, the router sends it back to the computer that requested it.  NAT
serves two functions though.  It also acts as a firewall.  It hides
all the LAN computers from the internet.  It does this by blocking port
requests, but also not making these computers visible.  It is a basic form
of a firewall, but security nonetheless.