Thanks, you've helped a ton. In response to your request:
The weight has the force of gravity acting on it, -mg. It, and heres the confusing part, also feels the tension in the string caused by the spool (mass M), which is moving in the -x direction, and is thereby translated to the positive y (along the string). So my''=Mx''-mg.
Another way to put it is The net force on the weight is the tension on the weight due to the spool minus the force of gravity on the weight.
My understanding of tension is that it acts in the opposite direction of the net force on an object, and acts along the line connecting said object to another object. So if the spool, mass M, were moving along positive x with an acceleration x'', the tension due to the spool would be Mx'' along negative x: -Mx''.