Which camera has better quality?

The Canon EOS 600D (T3i) or the Nikon D5100?
Also, if you know any other cameras you personally think is better, let me know.

Either or, pick whichever is most comfortable to you and has the features you want

You will get quality results from both as long as you know what you are doing.

Actually you can get rid of that word better when talking about these two companies and comparable cameras - they're both excellent, engineered to last many years.

Just pick one.

Everyone's response is correct. Try out each camera and get comfortable with using it to see for yourself. What you also want to consider is if you'll stick with Nikon or Canon down the line many years from now.

My coworker has the Canon 600D. It's really nice and light, sturdy, and takes great photos. The autofocus for video is great, while autofocus for photos lacks compared to the Nikon D5100.

My good friend owns the Nikon D5100.It was also light, sturdy, and takes great photos, except the Canon probably felt a little nicer to hold. Autofocus on photos is great, but definitely lacks in video autofocus compared to the Canon.

I've read there are some minor trade offs. RAW is better on the Nikon while JPEG is better on the Canon (whichever you choose to shoot).

There's only one reason why I wouldn't recommend the Nikon D5100, and that's because it is considered an "entry" level camera and therefore is not capable of autofocus on all of the Nikon lenses. It does not have a built in motor, so you will have to purchase AF-S lenses by Nikon for autofocus on this camera model. This was a move Nikon made to save some of the cost and weight of this camera. You may buy this and keep it with the one kit zoom lens for years and years, or you may eventually want to experiment with other lenses. As far as autofocus is concerned, you'll be limited on lens choices, though AF-S lenses are fantastic in image quality. Some people (like myself) have owned an entry level Nikon camera before upgrading to prosumer level DSLRs to use more lenses. This may or may not affect you depending on your usage/hobby.

For me, I recommended you to consider Canon EOS Rebel T3i. It's a light camera, easy to use, excellent image and video.

It's just a matter of preference. They're basically the same.