Webcam
With a simple, yet apt name, this product is an extension to the Apache web server which allows you to grab images from your webcam and upload them to your website. In order to configure and use Webcam and Apache you'll need working knowledge of servers (and configuration). Because of this I am hesitant to recommend the program to newcomers of Linux. However being a little bit tricky does not mean it is impossible. If you want to get your hands dirty, go for it!
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HasciiCam
If you want to unleash your inner geek and use your webcam to tweak and dabble with configurations, settings, imaging, etc. HasciiCam is for you. It grabs images from your webcam and saves them as ASCII files (text files) in your computer. You can set up a web page that refreshes at a predefined interval and use your ASCII files that it receives from your camera. Don't expect the crispest of output from this program but certainly expect fun results.
Image courtesy of Softpedia.
Geekast
If you are interested in Peercast, which is an open source multimedia multi-casting tool, then you would want to install Geekast, which is an interface to Peercast streams.
Conclusion
As discussed, there are a lot of programs to use for your webcam under Linux. Which one to pick? Well, it all depends on what you want to do with it. For fun photo-editing, Cheese and HasciiCam are the best options to go with, whereas ZoneMinder is an all-in-one surveillance solution for your home or office.
I've presented you with my preferred list, but there are many more options available. Search for “webcam” and “camera” in your package manager and check the results to find more webcam programs. Go exploring and have fun!