1. Voltages.
The mountain explanation where voltages are equivalent to height is a very good one. Let's pay a bit more attention to it....
If you have a circuit that is joined - such as the example given - then if you traverse from one point to the next, then the next, etc. until you get back to your starting point, then you have a net change in voltage of exactly zero. This is the exact same thing as traversing the mountain. You first pick a reference (eg up is positive and down is negative) and you have to keep using this same reference throughout your travels. Now whether the changes are positive or negative will depend on which way you decide to go. For example, if point 'x' is higher than point 'y' then if your journey goes from point 'x' to point 'y', then (using the previous reference) you will be travelling down - but if your journey takes you from point 'y' to point 'x', then you will be travelling up.
When applying this to a circuit, imagine yourself walking around it. Defining your reference direction between two points will have you connect the red lead of your DMM to one point and the black lead to the other. Let's say that as you walk between these two points, you will have the red lead in front of you and the black lead behind you - then you must always have the red lead in front of you and the black lead behind you. (The maths works exactly the same if they were the other way around, but once you pick a direction, you have to stick with it.)
If we look at your circuit, using the "red lead in front, black behind" reference, let's look at what happens when travelling from point 'a' to point 'b'. Firstly, the black lead will be on point 'a' and the red lead on point 'b'. There is a difference in potential of 4V with point 'a' being higher than point 'b' and with our "red lead in front" reference, the DMM will show -4V.
When you start walking from point 'b' to point 'c', you have to move both leads so that the red lead stays in front - which means the red lead will be on point 'c' and the black on point 'b'. The difference in potential here is 6V with point 'c' higher than point 'b' and with our "red lead in front" reference, the DMM will show +6V.
Using this same process of "red lead in front and black behind", when you go from 'c' to 'd' you will get a reading of +3V and from 'd' back to 'a' you will get -5V.
To prove you got this right, just add up all four measurements you took:
-4V
+6V
+3V
-5V
----
0V
Your answer should always be zero - since you are back where you started. This is Kirchhoff's Voltage Law.
Extension: Your circuit could have many other connections and components weaving in and around, but as long as everything was in steady state (Electrically this means DC) then you could do this exercise around ANY closed path and you will always get a total of zero.