I am looking at this Canon 50mm lens for my Nikon D5200. How can I be sure that this lens will fit on my Nikon D5200? The camera manual does not have a list of lenses that are compatible with the camera. Where can I find the information what lenses are compatible with my camera before purchasing the lens?
In general, you want to look at the lens mount. Cameras and lenses both use some standard of connector to ensure compatibility. In general, if a lens uses the same mount, it should be basically compatible, though some features may not work. For example, third party lenses may not support automatic adjustments on newer cameras without updates. Similarly, some older Nikon lenses can't focus on newer camera bodies that lack a focus motor built in to the camera.
The lens you have linked to is an EF lens which is one of two Canon mounts, so it would not be compatible with your Nikon which uses Nikon's F mount system.
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.)
The lens you've linked to is the Canon EF 50mm f/1.8 II. It will not fit your camera, because it is a Canon EOS lens: neither the physical linkage nor the electronic contacts match those on your camera. What you're looking for is a Nikon F-mount lens.
The easiest way to ensure you've got an F-mount lens is to buy a new lens that's Nikon-branded that is for dSLRs (i.e., not a Nikon 1 system CX lens). Any new FX or DX lens will be compatible with your camera. However, only lenses with AF-S in the lens name will have autofocus function on a D5x00 or D3x00 body. AF lenses won't autofocus. This is because AF lenses don't have a focus motor, and the D3x00/D5x00 bodies don't have a focus motor, and you need at least one motor to autofocus. If you upgrade to a D7x00 body, which does have a focus motor, then you can autofocus with both AF and AF-S lenses.
If you are shopping for third-party lenses from Sigma, Tokina, or Tamron, just make sure that you get the version that's Nikon-compatible, since those makers will create the same lens in multiple mounts. If you're shopping used, you may need to compare a picture of the lens mount with one of the Nikon F mount.
If you go vintage/used, and are looking at lenses that are more than 20 years old, you may want to get some additional advice on older F-mount lenses, which may or may not have electronic communication with the body (non-CPU lenses), and which may not be compatible with higher-end cameras, if you're planning on upgrading in the future (pre-AI lenses). These are still F-mount lenses, but can be problematic.
A couple other points.
The EF 50mm f/1.8 II is not a zoom lens. "Zoom" means that the lens has a variable focal length. The EF 50/1.8 II is a "prime" lens--the focal length is fixed.
You are probably looking for the closest Nikon analogs to the Canon EF 50mm f/1.8 II: the AF-S Nikkor 35/1.8G or AF-S Nikkor 50/1.8G. There is the cheaper (older design) AF Nikkor 50/1.8D, but it won't autofocus on a D5200.