discrete vs continuous random variables
Random variables come in two main flavours that you handle with completely different machinery. A discrete random variable can only take values you could in principle list one by one: 0, 1, 2, 3, ... such as the number of emails you get today or the count of heads in ten flips. A continuous random variable can take any value in a range with no gaps, like the exact height of a person, the time until a bus arrives, or a point chosen on a number line. There is no next value after 1.732 metres.
The split matters because of how probability is distributed. For a discrete variable, probability sits in lumps on individual values: each value can carry a positive probability, and those probabilities sum to 1. You describe it with a probability mass function (pmf), which gives P(X = x) for each possible x. For a continuous variable, no single exact value carries any probability at all (the chance of being exactly 1.732000... metres is zero); probability is spread smoothly along intervals, and you describe it with a probability density function (pdf), where probabilities are areas under a curve found by integration.
A useful test: can you, even in principle, list the possible values? If yes (possibly an infinite list like 0, 1, 2, ...), it is discrete; if the values fill an interval, it is continuous. Honest caveat: these two are not the entire story. Some variables are mixed, with a smooth part plus a lump or two (for example a rainfall amount that is exactly 0 with positive probability but otherwise spread continuously). The clean discrete-or-continuous picture is a starting point, not a law of nature.
Number of customers in a shop in an hour: discrete (0, 1, 2, ...). Time each customer waits in line: continuous (any non-negative real number). The same shop generates both kinds of variable.
Counts are usually discrete; measurements on a continuous scale are usually continuous.
Real measurements are recorded to finite precision, so they look discrete. We model them as continuous because the smooth pdf is far easier to work with and is an excellent approximation, not because reality has infinite resolution.