Which lenses are fully compatible with a Nikon D3200?

The kind of lenses you need depend a lot on what and how you like to shoot, and how limited you are by your budget. Basic features, like focal length, maximum aperture, and stabilization will be determined by your usage. See this basic guide to lenses for explanations of lens features and what they mean in practical terms.

You may also want to see Lens upgrade paths (sub $1000) for the EF-S 18-55mm IS kit lens for Canon APS-C cameras even though it's couched specifically for Canon, a lot of the lenses mentioned have parallels on the Nikon side of the fence.

But if you're looking at 3rd party lenses as well as those offered by Nikon, the main thing is that you need a lens in the Nikon F mount that is Ai or later. While you can use a pre-Ai lens with a D3200, if you eventually upgrade to a prosumer body, pre-Ai lenses can damage the camera body. With 3rd party makers like Sigma, Tokina, Tamron, or Samyang/Rokinon, "for Nikon" is probably sufficient. This means the lens will mount on the camera and work.

Because the D3200 is a "crop body" (i.e., has an APS-C sized sensor in it, not a "full-frame"), you can use either DX or FX lenses on it. DX lenses project an image circle that only covers an APS-C sensor; FX lens image circles are much bigger and can cover a full-frame sensor, too. You can use DX lenses on an FX body, but you'll get black corners in your images, or your camera will crop the image down to the APS-C area, at which point you've kind of negated the point of having a full-frame sensor.

If you want a lens to autofocus on your D3200, you also need to have a lens with a focus motor in it. In Nikon terminology, this means "AF-S", not "AF". 3rd party lenses will typically mention if they have the autofocus motor for Nikon or not.