An example of continuous is the number line. Between any two points, there are always more points.
For discrete sets, this is not true. For instance, in baseball there are four bases. If you get a hit it is either a one-base hit (what we call a single), a two-base hit, a three-base hit, or a home run. There is no such thing as a 2 1/2 base hit.
Discrete things are found in bundles or lumps, and you can only have certain numbers of them.
Money is another discrete idea because you can not sell anything for $0.005. Prices can be grouped for specials, like 2 for 99 cents, but if you buy one it is either 49 cents or 50 cents. Discrete does not mean it has to be whole numbers, but it does mean there are only some that can be chosen, and some can not.
Discrete sets can be infinite, but they can not be infinitely divisible. For example, the counting numbers from 1 to infinity are discrete, because, like the bases in baseball, you go from one to two and then to three but not the points in between. The number line from 0 to 1 is not discrete but continuous, because between any two points in the set, there is always another point. This is the key that makes the difference. In discrete we can talk about things that are "next to" each other, with nothing between them, while in continuous sets we cannot.