In general, all camera brands have their own proprietary system for connecting lenses. Modern mounts are all bayonet style, which means they twist and lock rather than needing to screw on, as older lenses did. These mounts are not interchangeable, so you'll need a lens that matches your camera bodies.

Most brands have different sub-variants of their mount, usually because they've been around so long and there have been changes over time.

Nikon's mount is called the F-mount, and it's one of the oldest. However, there are a lot of variations over the years and many older lenses are not compatible. Of more modern lenses, the main thing you'll need to watch out for is that your camera body doesn't have a motor to drive lens focus, so if you want to get autofocus, you'll need a lens which has its own motor. (This was a big concern when the first such Nikon bodies came out and you'll still see a lot of fretting over it, but these days there are many such lenses and especially at the entry level I wouldn't worry.) Nikon has an in-depth article about lens compatibility on their site: Which Nikkor Lens Type is Right for Your D-SLR?

Canon's mount is called the EF mount, and also has a variation called EF-S. The difference is that the EF-S lenses are designed only for APS-C cameras — the normal sensor size at entry- and mid-range levels now and for the forseeable future. More on this at What is the difference between EF and EF-S lenses? The EF mount is relatively young, and all lenses include their own focus motor.

Pentax's mount is the K-mount, and it's generally backwards compatible back half a century. However, newer bodies lack some mechanical couplings that older ones did, making some older lenses work only in limited form.

Sony's DSLR mount is called the A-mount, and actually comes from Sony's acquisition of Minolta. It also has pretty good old-lens compatibility.

And Sigma uses an incompatible variant of the K mount for lenses for their own camera bodies, but also makes models in many of the other mounts. There are a number of companies that make third party lenses in this way, either by reverse-engineering or by licensing agreements. In addition to Sigma, some of these are Tamron,Tokina, Samyang, and Zeiss.

Also, this is all for DSLRs. If we get into the mirrorless realm, there's a whole new set of letters — but the basic fact is that generally each brand has its own mount. (The Four Thirds consortium, which includes Panasonic and Olympus is the main exception; those brands share a mount, and there are also options from other names, including Kodak — but don't be fooled, because the real Kodak went bankrupt and this is a new company trading on the name, a strategy which hasn't resulted in any good products with names from similar classic companies like Polaroid or Vivitar.)