The short answer is Bonjour is a service (i.e. constantly running computer program) that runs on Mac OS X computers. It's purpose is to manage services and resources on your Local Area Network (LAN) with absolutely no intervention or configuration from you. Examples of services and resources include the following;
Access to files on other computers
Apple TV devices to playback movies on your big screen TV
Airport Express devices to playback music on stereo systems
Gaming computers and consoles that network for multi player games
The problem with short answers is they are not often clear and don't supply all the information you need to get what's actually going on. Feel free to read on and stop reading when your eyes glaze over. You won't hurt my feelings. LOL
Bonjour can be useful on home or small office networks that do not have a server to supply name look up help for the computers on the network. This is important because it is often the case that a user or a computer program on a given computer needs to use a service or resource that does not reside on that particular computer. The needed resource may however be available on another computer or device on the computer network.
When OS X on your computer is asked by you or by a computer program running on your computer to use one of these network services, it turns to Bonjour to find out the IP address of the computer or device that has the needed resource. Bonjour does this by reading a list of resources that it stores locally on your computer. This list is similar to a telephone book in that it keeps a list of device names and their corresponding IP addresses. But it also stores other information about what kinds of services or resources these devices have to share.
As Macs, printers and other devices come online on your network, each one advertises what resources it has to share to the other computers on the network. The Macs running Bonjour on the network record this information in their respective look up lists for later use.
If a needed resource isn't listed in the look up list, Bonjour broadcasts a query over the network asking for the resource being sought. Any computer or device that can offer the needed resource replies to the query with it's Name, IP address, the needed resource and any other shared services it has to offer. A session is then established to communicate between your computer and the remote computer or device that can offer the resource.
If you would like to know more, the following link is a good read about what Bonjour is and how it works. The article is targeted at software developers but it isn't too terribly technical and you might find it useful;
Hope this helps.