What is mg - T = ma ?

Could somebody please tell me in which instances do you derive which equation for physics?
My teacher did not reach this bit yet, but the homework questions requires the use of the equations (they're not in the text for some reason either). How do you change from F = ma into the two provided equations???

T = tension, g = 9.8 m/s^2, and etc...

mg - T = ma
T - mg = ma

F is the net force on the object. It is the total of all the forces - in this case, gravity and an upward-directed tension (maybe on a yoyo?).

T, g, and a all represent the magnitudes of vectors. You need to keep track of the direction of the vectors so that you know which ones to add and which ones to subtract.

If the object is accelerating downward, then make the downward force (gravity) positive and the upward force (tension) negative: F = mg - T
If the object is accelerating upward, then make the upward force positive and the downward force negative: F = T - mg

If that's too confusing, or if you don't know which direction the acceleration will have, then just pick a direction to be positive. A negative result will mean the acceleration is in the opposite direction.
(note: acceleration does not have to be in the same direction as velocity)

For example, suppose m = 10 and T = 50
If you decide that up is positive then T - mg = a, so a = -4.8 and the acceleration is downward.
If you decide that down is positive then mg - T = a, so a = 4.8 and the acceleration is still downward.

If the weight (mg) and the tension (T) are acting in opposite directions then the net force (F) is equal to mg - T. And F = ma, so ma = mg - T.