Estimate probabilities from its moments I want to estimate probability $Pr(X \leq a)$, where $X$ is a continuous random variable and $a$ is given, only based on some moments of $X$ (e.g., the first four moments, but without knowing its distribution type). 

As I had pointed out in my comments, it's hard to answer this question in generality. So, I'll just point you to a resource online.
But, that said, the magic words are generating functions-Probability generating functions and Moment Generating Functions.
The probability generating functions $\Phi_X$ exists only for non-negative integer valued random variables. The Moment generating function $M_X$ is related to the former [whenever and wherever both exist] by the following: $$M_X(t)=\Phi_X(e^t)$$
There are other inputs required, sometimes and sometimes not. So, please go through the material I have pointed you to.
EDITED TO ADD:
I'll get a little specific now:
If the random variable at hand has finite range, and you have $all$ the moments, then the distribution of $X$ can be found out, {Theorem 10.2, pp 5, 369 in the typeset}. If you just have first two moments, you'll get only Mean and Variance.
I'd love to hear from you incase you have specific queries. [Just add a comment below, I'll be notified!]
