For me, it helps to ask this question: Does it make sense to apply the midpoint? If it doesn't make sense, then the data is discrete. Otherwise, the data is continuous.
Here's an example:
Let's say we are counting the number of people buying a movie ticket. So the numbers we care about are positive whole numbers 1,2,3,... It doesn't make sense to apply the midpoint to the values 1 and 2 to get 1.5; you can't sell half a ticket. Therefore the variable "number of movie tickets sold" is a discrete variable. There are gaps in the data.
On the other hand, if we were measuring the weight of various coins, then this variable is continuous. It's possible to have any positive number decimal number weight (eg: 2.275 grams perhaps). It makes sense to apply the midpoint to any two weights and get some other weight. We don't have to worry about sticking to whole numbers only. Unlike before, there aren't any gaps and we consider this number set to be dense.
One way you could remember the two is to keep this in mind: If you count it, then it's likely discrete. If you can measure it, then it's continuous.
There are probably exceptions to that rule, but for the most part it holds up I think.
