What is a Real Number? I believe I'm over thinking it, but I want to be 100% sure.
A Real Number is any number, correct?  Whether it be an integer or something else.
It's the set $\mathbb R$ from $(-\infty, +\infty)$ correct?
 A: Yes, the integers and fractions are real numbers, as are things that cannot be expressed as fractions, such as $\sqrt 2$ and $\pi$. That is, the real numbers means the entire number line. On the other hand, $\infty$ and $-\infty$ are not real numbers.
The reason they are called "real numbers" as opposed to simply "numbers" is that higher mathematics works with various other kinds of things we call "numbers" too -- but those things do not correspond to points on the number line. You can safely ignore this until you need to learn about complex numbers (or transfinite ordinal numbers, or cardinal numbers, or p-adic numbers or whatever).
A: It seems most people are complicating the answer you are looking for.
Yes, you are correct, it is the set of all numbers on the number line, i.e any number in (-∞, +∞).
A: You can think of the set of real numbers as the set of all rational numbers closed under taking the limits of Cauchy sequences, i.e. all rational numbers plus all limits of Cauchy sequences you can build in the set.
However, not all numbers belong to the set of real numbers, see http://en.wikipedia.org/wiki/Complex_number. 
A: A common way to think about the real numbers as the set of every point on the number line between $-\infty$ and $+\infty$, including irrational numbers (like $\sqrt{2}$) and transcendental numbers (like $\pi$ and $e$).
For most people this is totally sufficient, though this doesn't really come close to telling you what they are, and it's circular anyway (how do you define the number line..?)
Perhaps a more illuminating way to think about the real numbers is as the rational numbers (i.e. the numbers of the form $p/q$ for $p,q$ integers and $q\neq 0$) with all of the 'gaps' filled in. In fact, that's the layman's description of one of the formal ways of specifying the real numbers -- as the completion of the rationals.
When we say that the rationals aren't complete, we mean that there are sequences of rational numbers that converge to (get closer and closer to) a number that isn't a rational. The sequence
$$\frac{1}{1}, \frac{14}{10},  \frac{141}{100}, \frac{1414}{1000}, \dots$$
is an example of such a sequence -- every number in the sequence is rational, but the limit is the irrational number $\sqrt{2}$.
This can't happen with the real numbers, because 'all the gaps are filled in'. Every sequence of real numbers that converges is guaranteed to converge to a real number.
A: Your definition is a bit loose...See this link for a more accurate definition and some examples:
Link-1
