Is self inductance and leakage inductance the same thing?
Let's start by defining the two terms:

Self-inductance is a measure or coefficient of self-induction in a circuit, usually measured in henries. It is also the property of an electric circuit that permits self-induction.

Leakage inductance is an inductive component present in a transformer that results from the imperfect magnetic linking of one winding to another.

Here is a longer definition of leakage inductance:


  Leakage inductance derives from the electrical property of an imperfectly-coupled transformer whereby each winding behaves as a self-inductance constant in series with the winding's respective ohmic resistance constant, these four winding constants also interacting with the transformer's mutual inductance constant. The winding self-inductance constant and associated leakage inductance is due to leakage flux not linking with all turns of each imperfectly-coupled winding.




It also might help you if you first understand what inductance is:

http://www.youtube.com/watch?v=84kl_WheNtQ

Here is another which is also very helpful:

http://www.youtube.com/watch?v=NgwXkUt3XxQ

Essentially, inductance is when the energy is stored in an electromagnetic field.



Now that you grasp inductance, let's look at self-inductance. 

Self-inductance is what happens when the current can change in a coil as a result of itself. It occurs as a result of the electromagnetic field (EMF), and has a relation to the proximity and shape of the coil predicting the scope and depth of the EMF.

Here is a video explaining self-inductance:

http://www.youtube.com/watch?v=LwwSJhEfJeA

Here is another video on self-inductance:

http://www.youtube.com/watch?v=B8CPGiK59f8

Even if you don't know the math, you should be able to glean the basic concept.

In a coil, the electromagnetic fields overlap. As a result, the overlapping EMF can feed itself (to an extent), and this is called self-inductance.



Leakage inductance, on the other hand, is a problem, not a property. Leakage inductance happens when self-inductance occurs undesirably, and by definition, leakage inductance will result in an unwanted outcome unless it is mitigated.

As a result of leakage inductance, what will happen is that the magnetic field does not follow the intended path. This will disrupt the flow of the EMF unless addressed.

Whereas self-inductance is a property of leakage inductance, leakage inductance is not a property of self-inductance.