Imagine two masses connected by a string draped over a bar.
Next, imagine that the blue bar (viewed head-on) is frictionless.  This means the string is free to slide over the bar with no resistance.
Your common sense should tell you that the red mass will accelerate DOWN and the green mass will accelerate UP.  We will call clockwise motion positive and counterclockwise motion negative.
Assume m2 > m1, so acceleration will be positive.

What forces act on m2?  Gravity and Tension in the string.

So, F2 = m2g - T = m2a  ("T" is negative because our acceleration is positive; "T" acts on m2 upward)

What forces act on m1?  Gravity and the tension in the string.

F1 = T - m1g = m1a  ("T" is positive becuse m1 is being raised up; "T" pulls m1 up against gravity)

The string that connects the two masses feels one force along its whole length, so we know that the Tension in both of these equations is the same.
It is also clear that the acceleration of both masses is equal, because they are connected by the same string.  In other words, a1=a2 (remember we said that a2 is positive).

If we put the two equations above together, and solve for T, we get...

T = m1g + m1a = m2g - m2a

They can also give us acceleration...

a = (m2 - m1)g / m1+m2

Login to MSU lecture OnLine.  Guests may use name=demo, pwd=demo, classs=brs.
Go to the chapter called "Newton" and look at the atwood applet.  It gives you a good feel for the topic, and it tells you the acceleration.