The adapter which carries this unique address may be a  NIC (Network Interface Card) or be built into the motherboard. The most common connection port type until very recently has been RJ45 but with  the increasing popularity of USB many broadband modems now connect to the computer via a USB port (which 'spoofs' a MAC address).

At one time broadband Internet Service Providers (ISP) would 'tie-in' their service to the MAC address of the user's computer. This could be very inconvenient when a user changed their computer/network card, or when they decided to run a local network behind a router. Fortunately this practice is being discontinued by most ISP's, but some users may still find they need to 'clone' the registered MAC address from the PC to their new router.

Secondly this is handled by the IP (Internet Protocol) address. There are many postings on the internet which will explain this in detail. In essence an IP address has the same function as a telephone number. The IP address may be 'static' (permanent) or 'dynamic' (subject to change). For example, if you connect to an ISP via a dial-up modem you will be given an IP address which will will exist only for the lifetime of the connection. Broadband users may be given either a static or dynamic IP address. In practice, a dynamic address may change only very infrequently. 

DHCP (Dynamic Host Configuration Protocol) is the method by which IP addresses are allocated and managed. Here is a typical set-up for a personal computer:

The key fact to note here is that the computer obtains its IP address automatically from the domain controller (DHCP) of the ISP. It also obtains the DNS server address(es) automatically (more about this later).

Here is a router setup screen (typically obtained by putting its IP address as the URL in a browser - http://192.168.8.1 in this case)

When it is 'looking outwards' the router is a DHCP client and obtains IP addresses - as shown above - from the ISP (what in effect your computer would normally receive). The subnet mask determines the number of addresses that can be allocated by a domain controller. Once again, you will find many postings on the internet which explain this in detail

Once the router is in place it has the task of acting as domain controller for its local network. It will allocate an IP address to each computer from its 'pool' - in the example above from 192.168.8.17 to 192.168.8.127.  These IP numbers are meaningless to computers outside the local network (which is the reason a router acts as an effective firewall).

Ping is one means of establishing whether a computer is operating on the internet. Here is an example:

In contrast the ping to this next IP address fails even though there are probably thousands of computers in the world which have exactly this address assigned by their local router:

So why does this next ping succeed ?

because it's a ping from a PC to a device on its local network (in this case a print-server)

If you choose to let the router act as a domain controller it will take on the job of assigning IP addresses to all attached computers and devices. You can however choose to switch off the router's DHCP and assign addresses manually as in this example:

There can be advantages to managing your own local IP addresses on a small network but generally it's more appropriate to leave it to the router. Note that the default gateway is the router itself. If the router's DHCP is switched off then the DNS server addresses supplied by the ISP must also be entered on each device requiring internet access.

DNS (domain name system/server/service) is the means by which IP addresses are converted into meaningful names. Click on this address  http://66.102.7.99 and you arrive at www.google.co.uk There are many postings on the internet which will explain in more or less detail how DNS works. Generally speaking users default to the DNS servers provided by their ISP but if you feel adventurous then you can augment that with your own - for example http://posadis.sourceforge.net/  Here is a quote from their web-pages

NAT (Network Address Translation)  describes the role of the router in making sure that traffic arrives at its proper destination. If PCs on the local network are sending information to and from each other, and/or are accessing pages from the internet, then this traffic must be routed correctly. Once again search on Google if you want more detailed information.

Networking is vastly more complex than described here but hopefully this has given you a flavour of what's involved when you set up your first local network.