Natural numbers (N) are 1, 2, 3, 4, ... (some authors include 0, some don't)
Integers (Z) also include the negatives: 0, 1, -1, 2, -2, 3, -3, 4, ....
Rational numbers (Q) are all the fractions, like 0, 1/2, 3/5, 4/1, 2/239, ...
Real numbers (R) are "all" numbers, so all the fractions and all the other (irrational) numbers like [itex]\sqrt{2}[/itex] and [itex]\pi[/itex]. If you want, you can think about a real number as any number which can be written in a (infinite, repeating or non-repeating) decimal expansion, like 1.23495012398530913298.......
(Sometimes, blackboard bold notation is used, for example [itex]\mathbb{N}, \mathbb{R}[/itex] instead of N, R.