Wireshark (formerly known as Ethereal) is a popular cross-platform packet sniffer --- it can record the packets that your network adapter sees on the network. Under normal circumstances, your adapter sees only the traffic that is sent to or from your computer. When you use a shared medium (such as a wireless network or a hub instead of a switch) you'll also be able to see packets belonging to other computers.
Packet sniffers such as wireshark and tcpdump are valuable tools to understand exactly what traffic is flowing on a link, and to debug network applications. However, in a non-switched environment it can enable you to see other people's traffic, and even in a switched environment you will be able to see other people's traffic if they are logged into the same machine. As a result, use of such tools should be determined carefully, and with respect to any AUP (Acceptable Use Policy)
Use wireshark to monitor your computer's traffic on the network. Since you likely don't have wireshark installed in /usr/sbin/wireshark, you'll need to install it first. Run yum install wireshark-gnome to install the gnome-based front-end to wireshark (say yes to install the packages requested and any dependencies).
Run wireshark. Note that this program requires root access (and so will ask you for the root password when started from a regular account). If you are unable to click on anything in wireshark after starting it, look for a hidden dialog box behind wireshark. Click on the Pseudo-device that captures on all interfaces --- it will start capturing traffic, and show the count of the number of ethernet frames captured of varying types. Stop after you've captured a few hundred and take a look at the kinds of traffic. If you aren't seeing any traffic, open a browser and start accessing some Web sites.
Use wireshark to capture just DNS packets (which are UDP packets on port 53). To do that, put "port 53" into the capture filter box (not display filter) that is visible from Capture->Options. Now visit a new site, such as http://www.cuny.edu and capture a few dozen DNS requests and responses.