I have a D5100 Nikon camera. Does the auto focus on the AF-SDX Micro Nikkor 40mm f/2.8g work on this camera. I have a micro Nikkor 55mm 1:28 lens which takes excellent pictures but can be difficult to use.

As Jennie has stated, yes but only in MF mode.

Nikon used to build the motor into the body for all their cameras but they started releasing lower end bodies without the AF motor to compete with Canon (their bodies don't have a focusing motor either). The result was that these lenses needed to have their own motor in them in order to be able to Auto-Focus.

This lens is the older D variant and does not have the focusing motor so you would need an enthusiast or pro body for this to work with autofocus as those bodies still have the focusing motor built in and yet they can still use the focusing motor in the lenses as well.

Yes all nikon (nikor) lenses will work with any of the D series nikons such as yours

works with camera in manual mode only

Use the 50mm 1.8 will work on the D90, it doesn't have an AF motor in the lens, but the D90 body itself has the motor built into it. Its a great lens and works perfectly on the D90

No. There is not a focusing motor in this lens so the autofocus will not work on the D90 body. It will work as a manual focus lens on that body, however. Consider going a little wider with the 35mm f/1.8 DX lens. Still has shallow depth of field and low light ability for a few more dollars.