I need to know if Nikkor 55 mm - 300 mm f/4.5 - 5.6 Super Telephoto Zoom Lens will work with my Nikon D5200 camera. Thanks.
Assuming you are referring to the Nikon AF-S DX NIKKOR 55-300mm f/4.5-5.6G ED VR - Yes of course it will work. The Nikon D5200 is a DX format camera with an F-Mount, and this is a DX format F-Mount lens. The D5200 does not have an autofocus motor built into it, but any lens that is "AF-S" has it built into the lens. So this lens will autofocus on the D5200.
It's an AF-S which has auto-focus built into the lens. Moreover, it's a DX lens which was built for your DX Nikon D5200. It'll work.
The Nikkor 55-300mm f/4.5-5.6G DX lens was a bit of a surprise when it was announced, as Nikon already had two very good telephoto zooms that appealed to DX body users (55-200mm DX and 70-300mm VR). Why Nikon was revisiting the low-cost telephoto space before filling out other DX options had a lot of Nikon users puzzled. Still, with two low-cost cameras (D3100, D5100) in the lineup, having low-cost lenses for them seems like the right thing to me. The question at hand is whether this was the right addition to the low cost lineup.
Yes, It will. DX means the same format your camera is made for. AF-s means the lens has an auto focus motor in it. as your camera does not have a built-in auto focus system, this lens will work perfectly.
However if you are too buried with the lens abbreviations, you can check out this link: http://www.kenrockwell.com/nikon/compatibility-lens.htm I found it pretty informative.