A discrete or dedicated Graphics Processing Unit (GPU) has its own RAM that's independent of the computer's RAM. For instance, the NVIDIA 8600M GT in my MacBook Pro has 512 MB of video RAM that's totally separate from its 4 GB of normal RAM. That 512 MB of RAM on the GPU is used only for graphics processing. Having its own dedicated RAM makes the GPU process graphics much faster, leading to increased performance.
One downside of dedicated GPUs is that they generate a great deal of heat. The GPU is often the hottest-running component in your Mac, especially if your Mac is portable. Another downside, which is more important in terms of performance, is that dedicated GPUs are very power-hungry, and they can drain a portable Mac's battery very quickly. Finally, dedicated GPUs also cost more than integrated GPUs, which drives up the cost of the entire computer; this is one reason why the MacBook has always had an integrated GPU rather than a discrete one.
Integrated GPUs don't have their own RAM. Instead, they utilize some of the system's RAM. The original MacBook's Intel GMA 950, for example, would use a minimum 80 MB of system RAM for graphics processing. The current MacBook's NVIDIA GeForce 9400M uses a minimum of 256 MB of system RAM. The upshots of integrated GPUs are that they're cheaper (meaning a less expensive computer), they put out much less heat (making them ideal for smaller Macs like the MacBook or MacBook Air), and they use far less power than a discrete unit.
For everyday graphics processing, video, Flash, and even 2D gaming, these integrated GPUs usually work well enough to get the job done. Where integrated GPUs fall flat is in 3D gaming. Because they don't have their own dedicated RAM, and because their processing units are generally much less powerful than those in dedicated GPUs, integrated graphics are usually considered suboptimal for 3D games.
All of this doesn't mean that you can't play 3D games on a Mac with an integrated GPU, but you probably wouldn't want to. Reports from Steam's forums indicate that users have been able to get Portal, a graphically-intensive 3D game, to run on an older MacBook's Intel GMA 950. However, the frame rate was generally only 9 to 15 frames per second (FPS). Ideally, you want a 3D game to be running at 30 FPS or more; if it runs at anything much less than 30 FPS, you're going to notice a great deal of stuttering and slowdown in the graphics. A frame rate of less than 15 FPS renders a 3D game pretty much unplayable.
More modern Macs' integrated GPUs, like the NVIDIA 9400M and NVIDIA GeForce 320M, will handle 3D gaming better than the MacBook's original Intel integrated graphics. However, you'll still most likely have to turn the games' graphics to lower settings in order to get a smooth frame rate.
So which Macs have integrated graphics?
-- All MacBooks
-- MacBook Air
-- 13" MacBook Pro
-- Mac mini (after the switch to Intel CPUs in early 2006)
-- Some models of 2006 17" iMacs
-- The 20" iMac (starting in early 2009)
-- Some configurations of the early 2009 24" iMac
-- Some configurations of the late 2009 21.5" iMac
The 15" and 17" unibody MacBook Pros also have integrated GPUs for use in power-saving mode, but they come with discrete GPUs in addition to the integrated graphics. Older unibody MacBook Pros required you to log out after switching between GPUs, but the newest models are able to do so automatically when needed. Pre-unibody MacBook Pros all came with discrete GPUs only. 27" iMacs and Mac Pros all have discrete GPUs, which are also available as a build-to-order option on the 21.5" iMac.
Unless you're into playing 3D games, integrated GPUs will almost certainly fulfill your graphics needs. If you want to play 3D games, though, Valve's assertion that integrated graphics aren't recommended for 3D games is generally correct. You might be able to play Portal on a current MacBook or MacBook Air, but in order to get a smooth frame rate, you'll probably have to turn the resolution and/or some of the graphics options down, leading to a less than ideal experience.