What type of lens fit Sony a580, canon 600d, 7D and nikon d1500?

And how would you know what type of lens fit other camera bodies?

Sony takes alpha mount. Canon takes ef and ef-s lenses. Nikon takes Nikon AF lens. Companies like sigma. And Tamron make lenses for all three.

Sony lenses fit sony cameras. Canon lenses fit canon cameras. Nikon lenses fit nikon cameras.

3rd party brands like Sigma and Tamron lenses have different mounts to accommodate each of these companies.

All camera manufacturers use a proprietary lens mount and there's no compatibility between brands.

The Sony A580 uses the legacy Minolta Maxxum "A" lens mount from 1985. That gives you access to a large number of good quality used Maxxum lenses.

The Canon 600D and all other Canon DSLR cameras use the "EF" lens mount first introduced around 1987. However, you must remember that there are now 2 lens mounts - the "EF" and the "EF-S". The "EF" lens mount lenses can be used on any Canon DSLR up to and including the full-frame 5D and 1D. The "EF-S" lens mount lenses can't be used on the 5D and 1D. So if someone buys a Canon T3i and 2 or 3 "EF-S" lenses if they ever upgrade to a 5D or 1D they'll have to buy all new lenses.

Nikon has been using their "F" lens mount since about 1960. Except for a couple of rare and exotic Nikkor lenses any Nikkor lens ever made can be used (with limitations) on any Nikon DSLR. Nikkor lenses designated as "DX" are designed specifically for the cropped sensor Nikon DSLR cameras such as the D40, D40x, D60, D3000, D3100, D5000, D5100 and D7000 but can still be used on a full-frame Nikon DSLR like the D700 or D3. The full-frame bodies simply crop their sensor to match the dimensions of the cropped sensor cameras. So you wouldn't be using all of the full-frame but at least you can use the "DX" lenses.

Tamron, Sigma and Tokina all offer lenses in brand-specific mounts so you aren't limited to just lenses offered by the camera manufacturer.

You know what fits, by checking that the lens mount is compatible with your camera.