BIRDS PHOTOGRAPHY With the NIKON D5000 camera?

I want to buy a NIKON D5000 camera for birds photography.
Which lens LENGTH will be good for that?

The new 55-300 mm may be just the lens for you.
You can save some money with the 55-200 mm ($200), but as a birder, you will love the extra reach the 55-300 mm ($400) will give you
If your budget is unlimited, of course the 400 mm f/2.8 or 600 mm f/4 would be amazing, but at $9, 550 and $13, 000, they get a little pricey

A 600 mm would be a good lens. Anything less will put you too close if you want a decent size image.

Best choice would be the new 600mm lens. That will cost about 10-15 times what the D5000 goes for. A more affordable choice would be a sigma 50-500 or 150-500 (make sure you get the HSM version)
Most affordable choice would be the new 55-300 but that will require you to get very close to the birds. I find that at 300mm I often have to crop and the more you crop the better your picture needs to be. Even though, it is possible at 300mm.
This is at my backyard feeder, 5 meters or so:
This was in the wild, 12 meters at least:
and quite a bit of cropping

If you want to be reasonable, a 55-200 or a 55-300 lens is enough. The 300 will let you zoom in more.
A 600mm f/4 isn't practical. First of all, it's around $10, 000, weighs 12 pounds, and you're going to probably need to shell out another $500 for a good tripod to support it.
I personally have next to no experience with bird photography so I can't comment much, however I have shot planes before and use long lenses for that.

Unless you have the budget to afford long prime lens such as a 400mm F4 or similar lens, I would recommend you consider the Sigma 50-500mm F4.5-6.3 APO DG OS HSM.
http://www.sigmaphoto.com/...-hsm-sigma
This lens is nicknamed the "Bigma" because of its size and focal range. It is a great lower cost alternative for bird photography. With the 1.5x crop factor of the Nikon D5000 it would have a 35mm equivalent focal range of 75-750mm giving you great "reach" to capture pictures of elusive birds.
I have one of these lenses for my Sony Alpha and love it. It is a great lens for the money and you can occasionally find them used on Ebay or Craigslist and save some money over a new one.

Sigma 150-500mm (HSM) is one of the best affordable lens to use on D5000 for birds photography.
One thing I have understood in birds photography - The longest lens that I own is always a little shorter and slower than what I need for that perfect bird shot.
It's the patience and common sense that counts and makes even a short lens get good results.

I would suggest you Nikon 55-300mm f/4.5-5.6G ED VR AF-S DX NIKKOR Lens
+ Excellent Image Quality
+ Solid build with metal mount
+ Significantly better in all ways than the Nikon 55-200mm VR
http://www.amazon.com/...B003ZSHNCC
http://rover.ebay.com/...m270.l1313