I am trying to understand the negative binomial distribution(also called gamma-Poisson distribution. What is the difference of it with the Poisson distribution anyway?), but it looks kind of similar with the binomial distribution that we study in basic statistics class. What is the basic difference between these two?
Both binomial and negative binomial distributions describe distribution of draws with replacement. The difference is the stopping rule that is used in both cases. Binomial distribution describes the number of successes $k$ achieved in $n$ trials, where probability of success is $p$. Negative binomial distribution describes the number of successes $k$ until observing $r$ failures (so any number of trials greater then $r$ is possible), where probability of success is $p$. So in the first case the number of trials if fixed, while in the second case it is random, but the number of failures is fixed. The without-replacement equivalents of the binomial and negative binomial distributions are the hypergeometric and negative hypergeometric distributions, respectively.
If you are looking to learn more about the probability distributions you can check the Statistics 110: Probability lectures by Joe Blitzstein from Harvard University that are freely available online.