Is there any difference between the free body diagram of fixed pulley and movable pulley? I've read that both of the rope of fixed pulley and movable pulley have the same direction (both upwards or downwards). But, one thing that confused me: is it true that fixed pulley has T1 and T2, but movable has T2 on both sides?
Is there any difference between the free body diagram of fixed pulley and movable pulley?
Not particularly. The main thing is that you can assume the fixed pulley isn't accelerating, so all forces on it must sum to zero. A movable pulley may or may not be accelerating.
is it true that fixed pulley has T1 and T2, but movable has T2 on both sides?
No. We can assume light strings have a uniform tension. This is because their mass is so small that any accelerations of a portion of the string contribute insignificant forces. Therefore the tension is solely due to the forces at each end of a span of the string.
The same principle applies to pulleys. If the pulley is light enough to be considered "massless", then any rotational acceleration of the pulley would contribute zero force to the string. Any difference in tension from one side to the other would be eliminated by the pulley rotating, so the tensions must be equal.
In contrast a real pulley has a non-zero moment of inertia. Whether fixed or moving, this resistance to rotational acceleration can allow a difference in tension from one side to the other.