Real Numbers
Real Numbers are just numbers like:
1  12.38  0.8625  3/4  √2  1998 
In fact:
Nearly any number you can think of is a Real Number
Real Numbers include:
Whole Numbers (like 0,1,2,3,4, etc)  
Rational Numbers (like 3/4, 0.125, 0.333..., 1.1, etc )  
Irrational Numbers (like π, √3, etc ) 
Real Numbers can also be positive, negative or zero.
So ... what is NOT a Real Number?
√1 (the square root of minus 1) is not a Real Number, it is an Imaginary Number  
Infinity is not a Real Number  
And there are also some special numbers that mathematicians play with that are not Real Numbers 
Why are they called "Real" Numbers?
Because they are not Imaginary Numbers.
The Real Numbers did not have a name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer!
The Real Number Line
The Real Number Line is like an actual geometric line.
A point is chosen on the line to be the "origin", points to the right will be positive, and points to the left will be negative.
A distance is chosen to be "1", and whole numbers can then be marked off: {1,2,3,...}, and also in the negative direction: {1,2,3, ...}
Any point on the line is a Real Number:
 The numbers could be rational (like 20/9)
 or irrational (like π)
But you won't find Infinity, or an Imaginary Number.
Real does not mean they are in the real world
They are not called "Real" because they show the value of something real.  

In mathematics we like our numbers pure, if we write 0.5 we mean exactly half, but in the real world half may not be exact (try cutting an apple exactly in half). 