This setup is an example of a Bernoulli experiment. A Bernoulli experiment is one in which the experiment is ran repeatedly and the outcome of each repetition can be either a success or a failure. Furthermore, success and failure occur with specific probabilities $p$. For example, we might say that on each repetition (also called a trial) the probability of a success is $p=.5$. It is also important that we specify the number of trials in advance. This just means that part of the specification of the experiment is that we will run, say, 10 trials.
If someone refers to a Bernoulli random variable what they mean is that the random variable counts the number of successes observed in the random experiment. For example, if you ran 10 trials and 5 of them came up as a success, this is the same thing as observing a value of 5 on the random variable.
So in answer to your first question, the random variable simply represents the number of successful kicks out of the total number of kicks taken.
The probability distribution of the number of successes in a Bernoulli experiment with parameters n (number of trials) and p (probability of success on each trial) is called a Binomial distribution:
$$
X \sim \textstyle {n \choose k}\, p^k (1-p)^{n-k}
$$
The above just says that your random variable $X$ follows the binomial distribution. The binomial distribution will attach an exact probability value to each possible observation of your random variable, given the supplied parameters. You can probably see that this function will produce a probability if you specify $n$, $k$, and $p$.
In the case in point here, the information that you know is $n$ and $k$ (because the coach is keeping measurements on this) and what you wish to know is the $p$. It is $p$ that really tells you something about the players' underlying ability to perform in the shootout. Having observations from their performance (i.e. $n$ and $k$) can help you estimate $p$.
