Prob in viewfinder in nikon d5100?

Whenever I try to take pic with viewfinder, it focusses but didn't clicks. Live view is working fine but don't
need that. When I focus some object even at well lit conditions too, it focuses and a green light in front of the camera comes up and then turns off.still dint clicks. Please refer me to any article or give a good answer. I'm an amateur photographer.

I'm not really sure what you mean by "it doesn't click"
If you're coming from a small digital camera(point and shoot) then take note that the focusing is different!
With my Canon 650D when i use autofocus, when it locks focus to something, a confirmation light blinks once! Same as you described it, a light turns on then off. That means that it has successfully focused. But if you press the shutter release button and it doesn't react, it means you should try to focus again, and if it takes the picture, if means that the focus was successful!

Http://keerok-photography.blogspot.com/2011/12/common-reasons-why-camera-wont-shoot.html

Is there a "?" flashing on the LCD or in the viewfinder? If so, then there's a button with "?" symbol (zoom out button). Press it and hold and see if it shows any message.

I own a Nikon D5100 too but have never seen a "green light" turning on at any time in any part of the camera (except the small LED, at the bottom-right of the back panel of the camera, turning on briefly when the image is being recorded/"written" onto the memory card)

By "Live view is working fine", do you mean you are able to just view the subject? Are you able to actually take photos with Live View? Have you previewed the pictures after taking them? If so, then the image capture mechanism is working fine.

If it shows "DEMO" mode, then either the memory card is not inserted (or is not compatible). Under such cases the shutter release is disabled (setting can be changed).

Do you get the red dot(s) in the view finder when you press the shutter button half-way? If yes, then focusing is working well too. Else, it could also be a case of low ambient light being insufficient for the camera to focus. At a point/area. Try switching both - the lens and camera modes to Manual and then release the shutter. (also try after turning on the flash).In lens manual mode, you can release the shutter even when it is not focused.

If it is unfathomable to find the root cause, take it to the nearest Nikon Service Center. They will check it and tell you the reason. If it is some genuine problem with the camera, be rest assured that it will be repaired or replaced with a new one.