There are many checks that can be done to verify that your network is working. Here is a non-exhaustive set of possible checks.
This is an IP (Internet protocol) address. It can best be thought of as your computer's cell phone number. When you connect to a server, or another computer, this is the identifier that allows the machine on the other side to respond. Normally, this is a behind the scenes thing. Most of the time you'll see a "hostname", such as www.yahoo.com. Hostnames are in turn, translated to IP addresses behind the scenes so we don't have to remember strings of long numbers.

In windows XP, select Start -> Run and type cmd.
Type ipconfig and you'll get a small bit of output, including the line "IP Address" with your machines IP Address.
Select Network Settings from Start->Settings.
Double click on the network profile you want to set the IP on.
This brings up the standard network diagnostics dialog. Hit the properties button.
This is the standard configuration panel. Select "Internet Protocol" and hit the Properties button.
Here you configure your IP by first hitting "Use the following IP address" if it is not already selected, then entering in the IP address, Netmask, and Default Gateway of your connection.
Hit OK, then OK on the next panel to commit the changes. Hit OK on any remaining panels.
If you get an error with the wording similar to "The DNS name server for the host specified in your URL could not be contacted. Please try your request again." first, check your typing. It may be a simple typing mistake. Here are a few other things you can try to verify the site is up.
Hit start->run and type cmd, then hit enter. Each of the steps below might help.
Type nslookup followed by the site name you are looking for, such as:
nslookup www.yahoo.com
This should return the ip address or addresses of the site you are trying to connect to. A response of "Server: Unknown" means something is not quite right in the DNS system somewhere. It could be an error on your machine or your ISP's DNS (can you get to other sites, if so, this is not likely the problem.) It could be the company hosting the site is having issues or the name has expired. It's best to try again in an hour or more in this instance.
Type ping site name and hit enter.
c:\>ping www.yahoo.com
Depending upon your ISP, this facility may be disabled, and could result in a failure. If it works, you will see the site name, followed by a set of numbers, like so:
c:\> ping localhost PING bradford73.com (127.0.0.1) from 127.0.0.1 : 56(84) bytes of data. 64 bytes from bradford73.com (127.0.0.1): icmp_seq=1 ttl=255 time=0.254 ms 64 bytes from bradford73.com (127.0.0.1): icmp_seq=2 ttl=255 time=0.036 ms 64 bytes from bradford73.com (127.0.0.1): icmp_seq=3 ttl=255 time=0.032 ms 64 bytes from bradford73.com (127.0.0.1): icmp_seq=4 ttl=255 time=0.037 ms
If you get this result, the site is up, but the particular service (web, email, etc) is unavailable. If this is at your ISP, you should contact your support options. Otherwise, it's best to just wait it out.
A hub works somewhat like a shortwave repeater. Any signal that comes in through any of its ports ( the receptacles the Ethernet wires plug into ) is sent back out to every other port in hopes of finding the correct machine. A switch works more like a cell phone call. The signal goes into the switch and only out to the specific port of the machine being contacted. Unless you cannot, you should always choose a switch. Modern networks are carry too much data for a hub to be an effective methodology anymore.
This can be difficult to do as some ISPs and network administrators will have this disabled. If it will work, you can do the following.
hit start->run and type cmd. Hit enter.
type tracert site name and hit enter
c:\> tracert www.yahoo.com
If it does work, you'll see output like:
c:\> traceroute www.yahoo.com 1 iocom-pnet-FA0-0.iocomcorp.net (206.224.79.77) 0.496 ms 0.368 ms 0.291 ms 2 pnet-gw.iocomcorp.net (206.224.79.82) 0.609 ms 0.462 ms 0.470 ms 3 hagg-01-t3-1-0-0-0.ausu.twtelecom.net (64.132.190.45) 1.263 ms 1.156 ms 1.134 ms 4 dist-01-ge-3-1-0-510.ausu.twtelecom.net (66.192.253.160) 1.396 ms 1.752 ms 1.159 ms 5 core-01-so-0-1-0-0.dlfw.twtelecom.net (168.215.53.93) 5.481 ms 5.465 ms 5.461 ms 6 66.192.255.85 (66.192.255.85) 6.062 ms 6.057 ms 5.716 ms 7 core-01-so-0-0-0-0.atln.twtelecom.net (66.192.255.25) 32.231 ms 32.161 ms 31.901 ms 8 core-02-ae0-1.atln.twtelecom.net (66.192.255.33) 61.529 ms 63.921 ms 31.917 ms 9 core-02-so-1-0-0-0.chrl.twtelecom.net (66.192.255.22) 43.733 ms 37.821 ms 37.313 ms 10 core-01-ae0-1.chrl.twtelecom.net (66.192.255.16) 37.323 ms 37.534 ms 37.318 ms 11 core-01-so-0-0-0-0.asbn.twtelecom.net (66.192.255.27) 46.624 ms 47.062 ms 46.614 ms 12 66.192.255.229 (66.192.255.229) 46.978 ms 46.412 ms 60.896 ms 13 exchange-cust1.ash.equinix.net (206.223.115.16) 47.152 ms 46.653 ms 46.984 ms 14 vlan200-msr1.dcn.yahoo.com (216.115.96.161) 46.800 ms 49.018 ms 46.930 ms 15 vl47.bas1-m.dcn.yahoo.com (216.109.120.218) 58.519 ms vl31.bas2-m.dcn.yahoo.com (216.109.120.146) 47.730 ms vl30.bas1-m.dcn.yahoo.com (216.109.120.142) 47.037 ms 16 p20.www.dcn.yahoo.com (216.109.117.207) 47.526 ms 47.332 ms 47.149 ms
This is a step-by-step path of how your network connection reaches the machine you are trying to get to. You'll see three numbers for each step. If you see asterisks, this indicates a network outage somewhere along the way. A router may be down or worse, a line has been cut somewhere along the way. If you encounter this kind of problem, and it is less than three lines from the top, you should speak to your ISP or local network administrators. If it is further down, most likely the network will learn how to route around the problem area shortly. Try again later in half an hour or so.
If you need to route between your home network and your internet connection, you can do the following.
Select your network settings from the Start->Settings menu. From there, select the network connection for your Internet connection.
It will pop up the standard network statistics dialog. Hit the Properties button.
The next thing you will see is the configuration panel for this network connection. We need to change a setting on the Advanced Tab.
Check the checkbox labeled "Allow other network users to connect through this computer's internet connection." Next, select another network connection profile in the pull-down labeled "Home networking connection." We selected Local Area Connection 2 in the example. Hit OK to commit the changes. Hit OK to close any remaining dialogs and you're done!