A traceroute test shows the path that a connection from your local computer to an online service takes.
When you connect to an online service (web site, email, ftp etc), there is no direct connection between your computer and the service you are connecting to. Your connection will travel through a number of different systems, pieces of hardware, and cables (both fibre and copper), when making the connection.
Traceroute information can be extremely useful because it can help to highlight problems on this route and show where the connection fails or the network slows down. For instance, if you are unable to connect to your web site you may assume that the server is unavailable, but if you carry out a traceroute the results may show you that you can't connect because there is a problem somewhere else on the internet and that the route to your web site is unavailable from your location. This would mean that the problem doesn't lie with our servers but with a issue on the internet that may be outwith our control.
If you ever have a problem connecting to any service, you should always carry out a traceroute to check whether there are network issues that are causing the problem you have.