Dirichlet Distribution - the underlying intuition.

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I'm not a math expert, but I need dealing with some math tools for natural language processing research. One of the most common tools is the Dirichlet distribution. I know that with a multinomial distribution I can estimate the probability of a vector of outcomes to happen given a vector of a priori probabilities.

I've tried to understand the Dirichlet distribution comparing to the multinomial case, but I'm facing difficulties since most "tutorials" use too much formal jargon, which I don't have enough background knowledge to absorb. My guess is that Dirichlet distribution is a multinomial with more "vectors" of parameters, but I'm not sure if it's the real thing.

Hopefully someone will be capable of bringing a proper answer according to my level of expertise in statistics (if there is any). Thank you in advance.