Which is a better lens for my Nikon D51 from usability perspective?

50-200 mm or 70-300 mm?

http://1.bp.blogspot.com/-elkYe2UVfZY/T5SLOKv5PEI/AAAAAAAAACs/CB3gOsey1rc/s1600/70-300mm lens.jpg

I'm guessing you really mean Nikon D5100.

As for he lens, it all depends what's important to you. If you have the kit lens, 18-55mm, then the 50-200mm will provide you seamless coverage from 18-200mm. But if a longer reach is desired, then the 70-300mm will be a better lens for you.

Though focal length is important, I really have no idea which model of the lens you're referring to. Some may be bad quality lenses.

If you genuinely have a Nikon D51 (and not a D5100) and you want to take photographs, then I would suggest that you sell the camera at a specialist camera auction, and buy either a D4 or D800 and a few lenses with the proceeds!

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Zoom Lens, Telephoto Lens, Macro Lens, Camera Lens, Wide Angle Lens, Cheap lens, DSLR lens

Neither, because the first is a Canon lens, and the second is Olympus. They are not adaptable to a Nikon DSLR body.

Neither.

The 50-200 in your photo is an Olympus E-mount lens and won't fit your Nikon (and I'm guessing you have a D5100). The 70-300mm is a Canon lens and won't fit either.

You can't mix or match lenses from one camera brand to another.

You must use Nikon lenses on a Nikon camera, or lenses from Tokina, Sigma, or Tamron (or others) specifically made for the Nikon F-Mount.

There are some junk 3rd party lens adapters available, but they are not very good.