Discrete logarithms on elliptic curves In many examples of attacks on public key cryptography, examples of the form $a ^ x = b$ are used, but I can not understand the correlation between this and the multiplication of the generator point of the elliptic curve $Q = xP$. 

I have already read a lot, but still I can not understand this. In the case $a ^ x = b$, this is one number, and for an elliptic curve it is a point with coordinates $(x, y)$. Please explain the difference simply, as you would to a child.
Well, to explain that, I have to touch lightly on what a group is, and how both operations involve a group.

A group [1] is a set of values, along with an operation (which I'll denote as $\odot$); with this operation, we take two values, smash them together, and generate a third one [2].  One example of a group are the numbers, along with the operation of addition $+$; we can take the values 1 and 3, and smash them together $1+3$ to get the value 4.

Now, one thing we can ask is 'if we start with a value $a$, how many $a$s do we need to smash together before we get the value we're looking for $b$'.  That is, if we compute:

$$\underbrace{a \odot a \odot a \odot … \odot a}_{\text{x times}}$$

how large does $x$ need to be before we get the value $b$?

This is called the discrete log problem.

With some groups, this is an easy problem.  For example, in the numbers and addition group I mentioned earlier, if we start with $7$, and ask "how many 7's do we need to add together before we get to 63, it's easy to see that the answer is 9.

Now, it turns out that both problems you mentioned are like this.

For the problem $a^x = b$, the $\odot$ operation is the act of taking two values, multiplying them together, and then taking the result modulo p (typically a large prime).  Because $a^x = \underbrace{a \times a \times a \times … \times a}_{\text{x times}}$, this is precisely the same form as the generic problem (and in fact is why we call the generic problem the 'discrete log' problem)

For Elliptic Curves and the problem $xP = Q$, the $\odot$ operation is taking two values where are 'elliptic curve points' [3], and performing an 'elliptic curve addition' to generate a third point.  That is, the problem is finding $x$ where $xP = \underbrace{P + P + P + … + P}_{\text{x times}}$ is the elliptic curve point $Q$; hence it is also in that same form.

Now, if the two problems are the same at the abstract level, why do we write them differently?  Well, it just comes down to notation.  For modular multiplication, we always write the operation as if it were multiplication, and so the obvious way to write a large number of multiplications is to write it as an exponentiation:

$$\underbrace{a \times a \times a \times … \times a}_{\text{x times}} = a^x$$

For Elliptic curves, we typically write the operation as if it were addition, and the obvious way to write a large number of additions is to write it as if we were doing a multiplication:

$$\underbrace{P + P + P + … + P}_{\text{x times}} = xP$$

This difference is just in the way we generally talk about these two grous - it has no impact on what we do with them.



[1]: That's a term that mathematicians use; don't worry, I would use that much more arcane terminology

[2]: Actually, there are a bunch of other things we require of a group - I will not mention them here.

[3]: I won't go into detail here about what an elliptic curve point is.