3rd Party Lenses
In this case, you are the 1st party, your camera maker is the 2nd party and so if you buy a lens made by someone else, they're the 3rd party!
The major 3rd party lens makers are Sigma, Tamron and Tokina. There are also some smaller companies catering for the lower end of the market which include Vivitar, Phoenix, Samyang, Quantaray, ProMaster etc. Many of these smaller "manufactures" really don't make lenses themselves but market generic OEM lenses with their name on them.
However Sigma, Tamron and Tokina do make and design lenses, some of which are of good optical quality and can compete head-to-head with lenses make by major manufacturers like Canon and Nikon.
The reason why people buy 3rd party lenses is usually price, though there are some focal length, speed or zoom ranges lenses are only made by 3rd party manufacturers, for example the Sigma 50-500/3.5-6.3 and 120-300/2.8 zooms.
The Canon EOS lens interface is quite complex being 100% electronic. The lens and body "talk" to each other via electrical signals and if they aren't speaking the same language there can be problems. Canon have said that they do not licence their lens technology to other manufacturers so the 3rd party lens makers have to "reverse engineer" the communications protocol. Generally Tokina and Tamron have done a pretty good job of this, with most of their lenses (past and present) work just fine with Canon EOS bodies.
Sigma, however, have a history of problems. Generally their current lenses work well with all current EOS bodies, but often when a new body is released, some of the older Sigma lenses have problems. If the lens is still in production Sigma will "rechip" it to work with the new body, but if it's out of production "rechiping" may not be offered. I don't know why Sigma lenses seem to have this issue, but in the past it's been a problem for quite a few Sigma lens users with EOS equipment.
In general, for Canon EOS users, I recommend sticking with Canon brand lenses unless there's a really good reason not to. With Canon lenses your risk of compatibility issues is zero and if there's some sort of problem, there's no doubt about who will fix it! You won't get into a situation where Canon blames the 3rd party and the 3rd party blames Canon. A good reason might either be saving a lot of money or if you want a lens that Canon doesn't offer. For example, the Sigma 50-500 is unique and nobody else, including Canon, has anything with that zoom range. So if you want 50-500 you have to go with Sigma (and by all accounts it's not a bad lens). An example of significant cost saving would be the Tamron 28-75/2.8, which is an excellent lens. Canon have a similar 24-70/2.8L, but it's 3x the price ($1100 vs. $370). If I needed a fast wide-to-normal zoom and I didn't want to spend more than $400, I wouldn't hesitate to go with the Tamron.