How to recognize what type of probability distribution to use in solving probability problems?

Step 1
The answer to this question is probably worth an entire book on probability theory, but I'll put in my two cents and point out some things about a few of those.
Bernoulli is a random "yes" or "no". In that case $X:\mathrm{\Omega <!--\; \Omega \; -->}\to <!--\; \to \; -->\{0,1\}$ with $\mathbb{P}[X=1]=\mathrm{\theta <!--\; \theta \; -->}\in <!--\; \in \; -->(0,1)$. Something happens or doesn't happen. This is useful in many situations, for example indicator functions (in analysis these are called characteristic functions) which simply tell you if something is true or not for some $\mathrm{\omega <!--\; \omega \; -->}\in <!--\; \in \; -->\mathrm{\Omega <!--\; \Omega \; -->}$ are Bernoulli distributed.
Binomial is the number of successes when you have several independent Bernoulli variables. Let $X_1,\dots <!--\; \dots \; -->,X_n$ be Bernoulli distributed with $\mathrm{\theta <!--\; \theta \; -->}=p\in <!--\; \in \; -->(0,1)$. Then the number of successes
$N=\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{X}_i$
follows a binomial distribution with parameters (n,p). This is useful e.g. in repeated experiments.
Poisson is the limiting case of binomial when $p_n/n\to <!--\; \to \; -->\mathrm{\lambda <!--\; \lambda \; -->}$ as $n\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$. It's easy to show using the characteristic functions of the random variables that such binomially distributed random variables N converge to a Poisson random variable with parameter $\mathrm{\lambda <!--\; \lambda \; -->}$ in distribution (weakly). Poisson distribution is an approximation of the binomial distribution when the sample size is large and the success probability is small.
Exponential distribution arises in Poisson processes: Assume that something happens at random times $t_i$. If the number of successes during an interval [0,T] follows a specific Poisson distribution (plus some independence assumptions probably) you have a Poisson process. The waiting time before the next success i.e. $t_{i+1}-<!--\; -\; -->t_i$ is exponentially distributed.
Step 2
Normal distribution is the limiting distribution of sample means (see central limit theorem) and hence arises naturally in several statistical applications. Student's t is most often encountered in statistical tests, for example the t-test statistic follows a Student's t distribution when the variance of the distribution is unknown. Chi square is also often seen in statistics, especially in "goodness-of-fit tests".
You probably realize that this is just scratching the very surface of the actual answer, which is worth of an entire course on this stuff. But here are some heuristics at least for some of those.