Despite Canon’s vigorous advertising campaign against third-party lenses a lot of people happily use lenses made by Tamron, Tokina and Sigma (and lenses with other brandnames but probably built by one of those three). And there’s one really good reason for this - the third party lenses are almost always much much cheaper than equivalent offerings from Canon.

So. Should you buy a third party lens? It’s not a simple yes/no issue, so here are some points to consider.

Price savings of third party products can be considerable, particularly if you’re looking to get a faster, higher-quality zoom lens.
Remember that the cheapest lenses are optimized for price, not for optical quality. And the profit margins for cheap products tend to be very thin. The price differential between Canon and third party isn’t huge when it comes to super-cheap lenses, so I don’t know if third party lenses are such a great idea in this case.

Third party makers produce lenses in a variety of market categories. Conventional wisdom is that if you’re considering third party at all you should consider the higher end of their product line, not the lower end for the reason above.
On the whole, Canon lenses seem to hold up their used value more than third party. If you intend to resell the lens anytime soon this can be a consideration.
Camera salespeople seem very eager to push third party lenses, so it’s likely that they receive bigger kickbacks from the manufacturers in return. Don’t let yourself be swayed by an eager salesperson - he or she probably isn’t trying to convince you to buy something for your benefit.
Buying Canon is pretty well a guarantee that your lens will work with any Canon EOS camera. However Tamron also have an excellent compatibility record with EOS cameras. Always test with your camera first, but be aware that the lens may not necessarily work with future EOS cameras.
Some older Sigma lenses do not work correctly with the latest EOS cameras. They fit the camera but don’t have compatible electronics, so the camera tends to lock up when you try to shoot. If you have such a lens you’ll need to contact the manufacturer to see if they can provide a free repair to the problem. The lenses don’t damage the camera - they just cause a temporary lockup that’s quickly cured by turning off the camera and turning it back on again.
Build quality of older Sigma products is notoriously inconsistent. A quick search of the Web reveals countless complaints from unhappy Sigma lens owners. Newer Sigma lenses seem to be a bit sturdier, judging by anecdotal evidence.
Many of Tokina’s lenses have heavy metal lens barrels, which take a lot of abuse but are a drag when hiking.
Canon offer many lenses with USM and full-time manual. Most third party lenses don’t have these features.
There are some operational differences. For instance, some third party lenses have focus or zoom rings which rotate in the opposite direction from the usual Canon direction.
It’s difficult finding useful comparative data. You can look up the MTF scores on sites such as Photodo, which is a useful guideline, but the only way to compare lenses properly is to test them yourself to see if they meet your needs. Asking, “Is the Tokina XYZ 2.8 lens better than the Canon XYZ 2.8 lens?” rarely yields helpful answers, because most people don’t buy both lenses and try them out.
Some specific third party lenses are better known than others. For example, Tamron’s 90mm macro lens has a reputation for excellent image quality at a price considerably less than Canon’s 100mm macro. Sigma sell an 8mm fisheye which Canon do not make.
But the biggest deciding factor is, as always, money. Only you can decide what’s your priority - low initial purchase price, mechanical reliability, compatibility, user interface or optical quality.