Can you put a different brand lens on a camera?

Of course you can. But there is the caveat that most manufacturers design their lens for their own cameras so you will have to look at secondary manufacturers who design lens to fit your camera. You will notice that these lens are either designed for Canon, or for Nikon, or for Olympus, etc. Camera lens are not as interchangeable as say car tires, where a Goodyear tire can fit on a Toyota or on a Honda.

Yes and no. There are many aftermarket manufacturers - Sigma, Tamron and Tokina being most notable - who make for the major camera brands. You will have to buy the proper mount, though, to match your camera.

Otherwise, there are some people who make adapters to mount one brand of lens on the "wrong" camera, but these are so full of compromise, that you are better sticking with the same brand or buying from one of the three that I just mentioned.

Nikon cameras need Nikon lenses or Nikon-mount lenses and so on...

if it is the same size and it is for that camera

First, 99% of all cameras sold do not have interchangeable lenses. Not only can they not take another lens, the main lens is not replaceable if damaged.

If you have a camera of the interchangeable type, then each brand has a different mounting base. I have a camera with the Pentax-K style, so it cannot accept a Nikon lens for example. Also features of the camera, such as auto-focus may not work with certain replacement lenses that do fit the same base.

And finally, lenses designed for 35mm film do not interchange gracefully with digital based cameras. That is because the image size on film is 1.5 inch, and the image size on the CCD is usually much smaller than 1 inch.