A discrete variable is one that is restricted (usually) to whole numbers or integers, while a continuous variable can take any real value.
Examples of discrete variables are frequencies of events, where the event can only happen a whole number of times. These variables often appear in statistics when associating a probability to a number of events occurring. They are also restricted to positive whole numbers.
Another example of discrete variables would be a problem in making up a sum of money using coins and notes. The solution would only contain discrete numbers of the various denominations of currency. They would be whole, positive integers.
An example of a continuous variable would be a graph of a relationship or an equation containing an independent variable and a dependent variable. Their relationship may be represented by a continuous curve.
Another example of a continuous variable would be temperature which varies continuously, even though it may be restricted to a range. Time is also regarded as continuous.
Anything which is countable is discrete. If it can't be counted, it's probably continuous.