1. Traceroute is a command built into most systems that traces the path through the Internet between two points (on Windows, it is known as "tracert"). Every IP packet on the Internet has a Time-to-Live field. Each router subtracts one from this field when it forwards the packet. This was designed to solve the problem of "routing loops": routers get confused as to the proper network topology and end up routing packets in an infinite loop. The TTL field guarantees that these packets will eventually die on their own accord. When a packet dies in this fashion, the last router to see the packet sends a diagnostic message back to the sender informing of this fact. Consequently (and this is the genius part), you can purposely create small TTLs that are guaranteed to kill the packet. You set the TTL field first to 1, causing the first router to drop it and inform you. You set the TTL field to 2, causing the second router to drop it. You do this for all TTL values, getting back a notification from each router in the network, thus mapping the route between two points. From Hacking-Lexicon

2. Traces the route taken by packets over a TCP/IP network. The traceroute utility displays the route used by IP packets on their way to a specified network (or Internet) host. Traceroute displays the IP number and host name (if possible) of the machines along the route taken by the packets. Traceroute is used as a network debugging tool. If you're having network connectivity problems, traceroute will show you where the trouble is coming from along the route. Install traceroute if you need a tool for diagnosing network connectivity problems. From Debian 3.0r0 APT

3. provides information concerning the route which packets must take to get from your computer (the server in this case) to a remote computer/server; typically used to diagnose possible problems in packet routing. From Linux Guide @FirstLinux