How to take an object and pause it in a moving video? I understand that this question doesn't make much sense, so allow me to explain. 

I have seen many videos, most of them involving sports and cars, where a person/car is frozen, and the object that was frozen is moving somewhere in the video will approach the frozen object and when it makes contact, the object that was frozen will now move with the object in real time/delete itself and the video keeps playing. 

I use After Effects and I wanted to know how this is done. If anyone can help, I'd really appreciate it.

I just spent an hour trying to find a video to at least give an example as to what I was talking about and I just can't find one, but I know I've seen it before.
In simple terms you split the video at the frame you want the objects to join up at, then also make a still image of the object at that frame (Layer 1 = video end of sequence, Layer 2 = still image Layer 3 = video beginning of sequence). If the camera is moving you will have to track the still object to match Layer 3. So the still image gets closer to the moving object as Video/Layer 3 plays, then when the objects join video/layer 1 cuts in and continues.