I am looking for a component that will take in a signal from another component, then tell another machine (motor) what to do. (It will tell the motor when to run, how long to run, and when to stop)

 

The list is way too long without some frame of reference.
Could even require a custom result!
Max.

 

It is called a MCU.

But no component is going to tell another device what to do unless you do the telling.

 

mcu, plc, pc

 

You have to tell us the details, otherwise you can't get the correct answer to match what you need.

 

There are a lot of vendors that will gladly sell you a drive that will do just exactly what you asked for. BUT given that you provided a really broad description and not much detail, it is probably not what you really want. But lots of drives are available for purchase. Providing an adequate description of what you seek is rather vital to the task of finding it.

 

How about a CNC Milling machine?
Max.

 

How about a CNC Milling machine?
Max.

Max, that only satisfies the very broadest part of the description, not the "motor" part of it.
I have used a programmed driver for a linear induction motor that did do exactly what was requested. The distance and rate were programmable and very repeatable. I will not wander off here and tell a story that is only distantly related to the topic.
But the short summary is that programmable motion controls are available from many sources and for almost any size of motor. But none of them are cheap.

 

Search this forum for "chicken coop door"

 

Max, that only satisfies the very broadest part of the description, not the "motor" part of it.
.

Considering the wealth of information in the OP.
I thought the emoji explained it.!

Supposed to be tongue-in-cheek!
Max.