Tension direction for pulleys in connected mass systems I got caught up with a conceptual question dealing with a practice problem with connected mass mechanics. 

Looking at the solution lecture notes is the following image for the Free Body diagrams for each mass, and the pulley, respectively:

Obviously, the Free Body diagrams on mass 1 and 2 are pretty straight-forward, but the situation with the pulley confuses me. $T_2$ going down is fine. The mass is falling, the system is overall causing $M_2$ to fall down, the string applies a force that way, sure. But $T_1$ confuses me. The mass is moving towards the pulley once we let the system act, so why is there a tension being applied that way? The directions for $T_1$ for mass 1 and the pulley are opposite for direction, and I'm not really sure why. Overall, the pulley is rotating clockwise, so this has to be the case -- $M_1$ will move towards the pulley, $M_2$ will drop. So why is $T_1$ facing that way? Intuitively speaking, that is.
 A: The force $T_1$ labelled $D$ in the diagram is the force on the pulley due to the string and it is trying to accelerate the pulley to the left, which is to be expected as the pulley is feeling a pull to the left from the mass $m_2$ communicated via the string.    
 
The force $T_1$ labelled $A$ in the diagram is the force on the mass $m_1$ due to the string and it is trying to accelerate the mass to the right, exactly as you expected.  
Perhaps what has confused you is the missing (trivial) FBD for the string with forces $T_1$, labelled $B$ and $C$ in the diagram, acting on the string.
In the context of the question those to forces are shown to be equal in magnitude because it is assumed that the mass of the light string is much less than the masses of $m_1$ and $m_2$.  
So the directions of all the forces labelled $T_1$ are consistent with the forces labelled $A$ and $B$ and $C$ and $D$ being Newton's third law pairs.

What is also missing from your FBDs is a force on the pulley pointing in a north-easterly direction.

Update as the result of some comments and questions.

Why does the mass of the light spring being much less than that of m1 and m2 show that the two forces are equal in magnitude? Also, is B and C both T1?

Perhaps I phrased that badly.
Assume that the magnitude of $T_{1\rm A} = 100\, \rm N$ and this force is the force accelerating mass $m_1$ to the right.
The magnitude of $T_{1\rm B}$ will be the same as the magnitude of $T_{1\rm A}$.
Suppose that the string had some mass but much less than $m_1$, say $\frac {m_1}{1000}$, then the magnitude of force $T_{1\rm C} = 100.1 \,\rm N$ to ensure that the string accelerates at the same rate as mass $m_1$.
So what I am saying is that the difference between $T_{1\rm C}$ and $T_{1\rm B}$ which is equal to $0.1 \,\rm N$ is much less that $100 \,\rm N$ and can be neglected.  

Additionally, what is causing this north-easterly directed force?  

The north-easterly force is the force that the support for the pulley exerts on the axle of the pulley.
If you did not have that force there then the resultant of forces $T_{1\rm D}$ and $T_1$ is a south-westerly force which would cause the pulley to accelerate in that direction.  

Also, if my third law force pairs thinking is correct, then why is that not applied to T2 on the pulley? It pulls down on the pulley, just as the mass goes down.

The downward force $T_2$ acting on the pulley is due to the vertical part of the string.
The upward force $T_2$ acting on mass $m_2$ is due to the vertical part of the string.
As drawn these are no a Newton third law pair.
The FBD is the vertical string is missing.
Very often when such FBDs are drawn the FBDs for the strings are omitted because they do not add much to the actual calculation in hand.
A: Intuitively speaking, the mass $m_1$ is decreasing the speed. So it must reduce the speed of the system by exerting a force on the pulley that opposes that direction of movement. So $T1$ must be leftwards.
A: Strings cannot push unlike a coiled spring.Hence if we show the forces acting on a point due to a string the tension has to be pulling that point not pushing it.
