Variable - A variable is a quantity which we do not yet know the value of and is denoted by a lower case letter e.g. total number of sales this week at a newsagent, gender of an employee, the annual rate of inflation etc.
Random Variable - A variable whose value depends on the outcome of an experiment. For example, our variable could be the outcome of flipping a coin. There are two outcomes, heads or tails, and the probability for each outcome is $\dfrac{1}{2}$.
Note: Capital letters $X$, $X_1$, $X_2$, $Y$ etc. are used to denote random variables, and lower case letters $x$, $x_1$, $x_2$, $y$ etc. are used to denote the corresponding values (i.e., the actual values that the random variables take).
Discrete Random Variable - A variable which can take one of a countable set of values within a specified range. Examples include the number of students present in a class (a student can't be half present!), a student's grade level, prices etc..
Continuous Random Variable - A random variable which can take any value (not just whole numbers) within a given range, e.g. annual rate of inflation, the height of a student, time etc..
Probability Distribution - A probability distributions tell us all the values a random variable can take, along with their associated probabilities. A probability distribution can be presented as an equation, a chart or a table. For any probability distribution, the sum of the probabilities of all mutually exclusive events is always one.
Expectation - The average (mean) value of a random variable. Or more formally, the value of a random variable we would “expect” to see if we repeated the random variable process a very large number of times.
Variance - The variance tells us how spread out the values of the outcomes of an experiment are. Equivalently, it tells us how far away, on average, the values of the outcomes of an experiment are from the mean value. A large variance indicates that the values will very spread out, whereas a small one suggests the values lie close to the mean.
To develop these ideas further see the pages on discrete probability distributions and continuous probability distributions.