If I buy an AF-D lens, the Sigma 70-300mm for Nikon D3200, from Amazon, will it have an autofocus motor?
There's a listing with "Built in Motor for Nikon". What does this mean?
Possibly so, but not necessarily. That is Not what AF-D means. AF-D means the lens reports focus rotation as the subject distance. All Nikon lenses since 1992 are D lenses, regardless if they say so or not. Lenses that have more to say (like AF-S) will say that instead. AF-D can mean that it has nothing else to be said.
The Sigma 70-300 says it does have the motor and will auto-focus on on all bodies (read this for yourself, don't trust me, I don't have that lens), but many AF-D do not have the lens motor.
See the lens descriptions at http://www.bhphotovideo.com (best descriptions around). They normally say it will or will not autofocus on cameras like D40 or D60 (which have no motor in the camera body, and thus require the motor in the lens).
The lenses that say AF-S do have the motor in the lens. Technically, AF-S means it has the newer ultrasound silent motor, but it also means it has a motor.
The concept is that Nikon bodies used to have the motor in the body, and focused the lens with a screw drive. Then AF-S was invented, nicer motor in the lens driven electrically. Then Nikon started omitting the motor from the body, since the lens had one. The bigger bodies (D300, D800, etc) still have the body motor too (can use older lenses), but the lesser bodies omit the motor now. AF-S lenses will have the lens motor. Others might too, but you have check them.