Variables in statistics are a lot like variables in algebra or calculus, except that they are typically thought of as having a distribution of values. For instance, a variable might describe someone's reaction time to a stimulus, but this variable it will be different each time it is measured. So a variable like reaction time can be understood as a distribution---a collection of all possible reaction times and how likely each is---rather than a single value.
Variables in experiments can be continuous or discrete. In psychology experiments, continuous variables are almost always real numbers (e.g. people's height, reaction time). Discrete variables are ones that do not vary continuously. Typical examples in psychology experiments take on only a finite number of values, for instance in measures like level of education, number of children, or correct/incorrect. However, it is possible that discrete variables give a potentially infinite number of values, as in counts of how neuron spikes occur in a given amount of time.