Wikipedia has a page that lists many probability distributions with links to more detail about each distribution. You can look through the list and follow the links to get a better feel for the types of applications that the different distributions are commonly used for.
Just remember that these distributions are used to model reality and as Box said: "all models are wrong, some models are useful".
Here are some of the common distributions and some of the reasons that they are useful:
Normal: This is useful for looking at means and other linear combinations (e.g. regression coefficients) because of the CLT. Related to that is if something is known to arise due to additive effects of many different small causes then the normal may be a reasonable distribution: for example, many biological measures are the result of multiple genes and multiple environmental factors and therefor are often approximately normal.
Gamma: Right skewed and useful for things with a natural minimum at 0. Commonly used for elapsed times and some financial variables.
Exponential: special case of the Gamma. It is memoryless and scales easily.
Chi-squared ($\chi^2$): special case of the Gamma. Arises as sum of squared normal variables (so used for variances).
Beta: Defined between 0 and 1 (but could be transformed to be between other values), useful for proportions or other quantities that must be between 0 and 1.
Binomial: How many "successes" out of a given number of independent trials with same probability of "success".
Poisson: Common for counts. Nice properties that if the number of events in a period of time or area follows a Poisson, then the number in twice the time or area still follows the Poisson (with twice the mean): this works for adding Poissons or scaling with values other than 2.
Note that if events occur over time and the time between occurrences follows an exponential then the number that occur in a time period follows a Poisson.
Negative Binomial: Counts with minimum 0 (or other value depending on which version) and no upper bound. Conceptually it is the number of "failures" before k "successes". The negative binomial is also a mixture of Poisson variables whose means come from a gamma distribution.
Geometric: special case for negative binomial where it is the number of "failures" before the 1st "success". If you truncate (round down) an exponential variable to make it discrete, the result is geometric.