I want to purchase a camera with a built in autofocus motor, but I want to know if lenses without autofocus will then focus automatically on that camera body.
In other words, can I buy a Nikon camera body that already has autofocus, then purchase any non-autofocus Nikon lens and be able autofocus simply from the body itself?
Actually, it's kind of the opposite. You need a lens with autofocusing capabilities--you just don't need one with a focus motor. That is, you'd still need AF lenses, they just don't have to be AF-S or AF-I lenses. AI lenses, for example, were designed before Nikon added autofocus to its cameras and lenses, and can only be manually focused, regardless of whether the camera body has a focus motor or not.
However, these AI lenses were only made by Nikon until 1986. So, if you're looking at purchasing a new lens, it's unlikely you'll run across one. There are still some manual lenses in the Nikon lineup, but none of them have the AF or AF-S designation.
No — an autofocus motor in the body will not make older, manual focus lenses into autofocus lenses.
Any autofocus lens needs a motor — it's what does the "auto", after all. (Just like an automobile would be just a ... moble without one.) There are two primary places where this motor can be placed — either in the lens itself, or in the camera body. Each of these approaches has advantages and disadvantages, but that's beyond the scope of this question.
The important thing is that in-body motors need to deliver their output (in the form of motion) to the lens in some way. In order to do that, the lens needs to be designed for that connection. Usually, this is done via an autofocus screw — a drive shaft that connects from the body to the lens, and lenses designed to be focused in this way are sometimes called screw-drive lenses.
Now, historically, there's an interesting footnote, because one early autofocus system, the Contax AX worked without a connection to the lens at all. Instead, it moved the film plane within the camera body. That meant it could work with lenses not designed for it. But, this approach had many limitations (detailed in the linked article, if you're curious), and that's why it's just a historical footnote — it doesn't apply to Nikon or to any modern camera.
Also see What is an auto-focus motor? for more.
(Good, thorough answers here already, but I'd like to add a shorter, simplified one...)
A camera body with autofocus capability is not magically able to autofocus with any lens that can be mounted. The only lenses that can be autofocused are lenses that have been specifically designed for autofocus.
Nikon camera bodies with an internal focus motor allow you to use autofocus with the (mainly older) AF lenses that don't have a focus motor in the lens.
There's basically a spring loaded screwdriver style blade driven by the body in the motor that engages with a socket in AF lenses without a motor, that drives the focus mechanism in the AF Lens.
With a manual focus lens, there's no socket so the body drive connector stays pushed back inside the mount. An MF lens remains an MF lens - there's no magic to make it autofocus, it needs the AF focusing mechanism in the lens to work. (Similarly, an AF lens that needs the body motor won't autofocus with a body that doesn't have the focus motor).
However, there used to be a way to get partial autofocus with some Manual focus lenses and a body with the focus motor - Nikon used to do a teleconverter (I think it was the TC-16A) that accepted some manual focus lenses, and (for bodies with the body focus motor) provided limited autofocus by moving the telecomverter lens elements. It wouldn't give you full range AF, but if you got the focus in roughly the right range, it would tweak it to get it in focus. However, this was produced quite a few years back (I think it was around the time Nikon started doing AF lenses) so you'd have to look at the used market - and I'm not sure whether it's compatible with current bodies.
To be clear, the older Nikon lenses, which are no longer manufactured but are widely available on the second hand market, were manual focus lenses. That is, they did not have any kind of autofocus capability regardless of what camera body you use them on. If you get a Nikon camera body with a built in autofocus motor, you will still be able to use these old manual focus lenses BUT you will have to focus them manually.
The point of having a Nikon body with a built in autofocus motor is that you will be able to use it with some of the older AUTOFOCUS lenses which do not have motors built in. These older autofocus lenses are usually called "AF-G" lenses. All modern Nikon lenses which currently being manufactured have built in motors and are designated as "AF-S" lenses. (The "S" stands for "silent-wave" - Nikon's name for the nifty little motors they use in their lenses).
So the only reason why you would specifically need to buy a Nikon body with a built-in motor is if you wanted to use it with AF-G lenses. If you want to use the old manual focus lenses, or the current range of AF-S lenses, then you don't really need a Nikon body with a built-in motor.
If you need more information you can check out this page on the Nikon site.
Good luck!
