What is the lens for a DSLR camera used for?

Is it used for zooming in or focusing while taking a picture or both? If both, do all types of DLSR lenses focus and zoom? What does the lens on the Nikon D3200 do? (focus or zoom or both?)

The strength of any lens, normally written in millimeters (e.g. 50mm) is referred to as the focal length. A zoom lens like an 18-55mm lens has an adjustable focal length of 18mm to 55mm and everything in between.
The other type of lens is what's called a prime lens. Prime lenses do not zoom as they have one, and only one focal length. Examples: 24mm, 50mm, 85mm, 105mm are all prime lenses. You can have wide angle prime or telephoto primes.
All lenses for DSLRs, regardless if they prime or zoom, will have a focusing system. The system may vary from camera to camera, but all lenses for DSLRs have a focusing system. The only time you'll find a lens that does not focus is on large format cameras like the type used by Ansel Adams. Here's an example of modern one: (http://www.bhphotovideo.com/c/product/63358-REG/Wista_214604_45DX_Rosewood_4x5_Wooden.html
As you can see in the photo, these camera have frame to hold the lens, and they focus by moving that frame on rail system. Very old fashioned and low tech for sure!

Then there's this guy: http://www.bhphotovideo.com/...amera.html
Like the wooden camara, this Mamiya has the same kind of focusing system where the lens fits onto a frame and the frame is moved instead of moving a ring on the lens. This allows lenses to have simpler, cheaper and sometimes better optical designs.

Can you focus or zoom while taking a picture? Yes you can. If you use a very long shutter speed like, say, 5 seconds, you can turn either the zoom or focusing to achieve various special effects. You need a long shutter speed to give you time to turn the focusing or zoom ring enough to have any kind of effect.
Here's an example of what happens when you turn the focusing during an exposure: https://500px.com/...hotography
And here's what happens if you zoom during the exposure: https://500px.com/...tim-drivas

First of all, if a lens can't focus it is useless.

Zoom lenses allow the user to frame their shots without needing walk closer or further away from the subject.

I'm sure with is information, you can answer your question yourself.

Below is a link to all the Nikkor lenses made at this time. The lenses you want to choose from are the Nikkor AF-S lenses (zoom or prime)

At the most basic a lens is used to create an image on a plane of a view that is in front of it. It does that by focusing the image onto the plane - in the case of a digital camera the plane is the sensor, in a film camera it's the film. Without a lens no image forms as light from the view would be spread across the image plane. A lens does not have to be transparent glass, it can just be a small hole or even a finely printed set of concentric circles (commonly called a zone plate).

So to be of any use a lens, no matter what it's construction has to be able to focus an image, that is all that is needed. When it comes to DSLR lenses, many, called "primes" do not zoom", their advantage being that removing the zoom ability tends to make for a better quality lens.

It would be possible to construct a lens that not only had no zoom, but also had no adjustable focus. To use such a lens you would have to place the object you wanted to photograph (or view) at the correct distance from the lens before you'd get a sharp image on the image plane!

Both. A zoom lens zooms and focuses. As you can read into what I said, there are some DSLR lenses that are prime lenses which means they do not zoom, but, of course, they do focus. The Nikon D3200 is the body of a DSLR. One put various lenses on it, including the kit lens, an 18-55mm zoom lens, that, yes, does focus and zoom. But, like all DSLR's, you can also put on a prime lens, such as the 35mm f/1.8 lens; which does not zoom, but does focus.