Atwood's machine is a device where two masses, M and m, are connected by a string passing over a pulley. Assume that M > m.
What is the acceleration of the two masses?
Start with a good freebody diagram. Two, in fact, one for each mass.
Assume the pulley is frictionless and massless, which means the tension is the same everywhere in the string. We'll learn how to account for the pulley later in the course.
T_{1} = T_{2} = T
Think about what the system will do. If the system is released from rest, the heavy mass will accelerate down and the lighter one will accelerate up.
Align the coordinate systems with the acceleration. Each mass has its own coordinate system, but they must be consistent.
Take +y down for mass M.
Take +y up for mass m.
Recognize that the masses have the same acceleration, a.
Apply Newton's second law for each mass.
For mass M:    For mass m:  
SF_{y} = Ma_{y}    SF_{y} = ma_{y}  
Mg  T = Ma    T  mg = ma 
Combine the equations to eliminate T. T = mg + ma, so:
Mg  mg  ma = Ma
a (M + m) = g (M  m)
a  = 

g 
Check your answer by plugging in some limiting cases:
Numerical example: M = 210 g and m = 200 g.
a  = 

g  =  0.24 m/s^{2} 