Is the voltage between the two points $A$ and $B$ denoted as $U_{AB}$ or $U_{BA}$? And why? Consider the following circuit 

Is the voltage between the two points $A$ and $B$ denoted as $U_{AB}$ or $U_{BA}$? And why?
 A: It is all down to the convention which you are using.
$U_{\rm ab}$ could mean the potential of node $a$ relative to node $b$ or the potential of node $b$ relative to node $a$.  
I think that more often "potential of node $a$ relative to node $b$" is chosen?
For example when dealing with transistors $U_{be}$ is taken to mean the potential of the base relative to the emitter.  
In your diagram with the node to the left of the capacitor labelled $A$ and that to the right of the capacitor labelled $B$ and with the arrow direction by the capacitor shown from right to left I would say that the potential difference across the capacitor labelled $u_{\rm C}$ could be labelled $U_{\rm AB}$.  
If the node at the bottom of the resistor was labelled $D$ then one might use $U_{\rm BD}\:(=u_{\rm R})$ as the potential difference across the resistor.  
You would then have the potential difference across the capacitor and resistor $U_{\rm AD} = U_{\rm AB} +U_{\rm BD}$.  
Ignoring the ammeter then the potential difference across the voltage source is also $U_{\rm AD}\:(=E)$ 
With the direction of the current as shown in your diagram it looks as though the passive sign convention is being used and so application of Kirchhoff's voltage law produces the following equation $E - u_{\rm C} - u_{\rm R} = 0$ which you will note is the same as $U_{\rm AD} - U_{\rm AB} -U_{\rm BD} = 0$.  
A: I agree with @Farcher’s answer. Here’s a slightly different way of looking at it. 
The definition of voltage or potential difference is:
The potential difference $V$ between two points is the work (energy) per unit charge required to move the charge between the two points.
Work is a scalar quantity so it doesn’t have a “direction”. So it doesn’t matter if you say $U_{AB}$ or $U_{BA}$. $U_{AB}$ can mean the potential of node A relative to node B or vice-versa.  The electric field between A and B, on the other hand, is a vector quantity and therefore has direction which, by convention, is the direction of the force that a positive charge would experience when placed in the field.  That would mean the field is from A to B.
With respect to the application of KVL, by convention, the direction of current is the direction of movement of positive charge. So if positive charge moves from plus to minus in a field, it losses potential energy, and therefore experiences a drop in voltage (potential) and is assigned a negative value.  For the current direction shown in the diagram the charge experiences a drop in voltage in going from A to B and from B to the node on the other side of the resistor. The difference is energy is stored in the electric field of the capacitor but dissipated as heat in the resistor. 
Hope this helps.
