What is Infinity?
Infinity ...  
... it's not big ...  
... it's not huge ...  
... it's not tremendously large ...  
... it's not extremely humongously enormous ...  
... it's ... 
Infinity has no end
Infinity is the idea of something that has no end.
In our world we don't have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.
So don't think like that (it just hurts your brain!). Just think "endless", or "boundless".
If there is no reason something should stop, then it is infinite.
Infinity does not grow
Infinity is not "getting larger", it is already fully formed.
Sometimes people (including me) say it "goes on and on" which sounds like it is growing somehow. But infinity does not do anything, it just is.
Infinity is not a real number
Infinity is not a real number, it is an idea. An idea of something without an end.
Infinity cannot be measured.
Even these faraway galaxies can't compete with infinity.
Infinity is Simple
Yes! It is actually simpler than things which do have an end. Because when something has an end, we have to define where that end is.
Example: in Geometry a "Line" has infinite length ... it goes in both directions without an end. If it has one end it is called a Ray, and if it has two ends it is called a Line Segment, but that needs extra information to define where the ends are. 
More Examples: 

{1, 2, 3, ...} 
The sequence of natural numbers never ends, and is infinite. 
OK, ^{1}/_{3} is a finite number (it is not infinite). But written as a decimal number the digit 3 repeats forever (we say "0.3 repeating"): 0.3333333... (etc) There's no reason why the 3s should ever stop: they repeat infinitely. 

0.999...  So, when we see a number like "0.999..." (i.e. a decimal number with an infinite series of 9s), there is no end to the number of 9s. You cannot say "but what happens if it ends in an 8?", because it simply does not end. (This is why 0.999... equals 1). 
AAAA... 
An infinite series of "A"s followed by a "B" would NEVER have a "B". 
There are infinite points in a line. Even a short line segment has infinite points. 
Big Numbers
There are some really impressively big numbers.
A Googol is 1 followed by one hundred zeros (10^{100}) :
10,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000
A Googol is already bigger than the number of elementary particles in the known Universe, but then there is the Googolplex. It is 1 followed by Googol zeros. I can't even write down the number, because there is not enough matter in the universe to form all the zeros:
10,000,000,000,000,000,000,000,000,000,000,000,000, ... (Googol number of Zeros)
And there are even larger numbers that need to use "Power Towers" to write them down.
For example, a Googolplex can be written as this power tower:
That is ten to the power of (10 to the power of 100),
But imagine an even bigger number like
And we can easily create much larger numbers than those!
Finite
All of these numbers are "finite", we could eventually "get there".
But none of these numbers are even close to infinity. Because they are finite, and infinity is ... not finite!
Using Infinity
We can sometimes use infinity like it is a number, but infinity does not behave like a real number.
To help you understand, think "endless" whenever you see the infinity symbol "∞":
For example: ∞ + 1 = ∞
Which says that infinity plus one is still equal to infinity.
When something is already endless, we can add 1 and it is stil endless.
The most important thing about infinity is that:
∞ < x < ∞ Where x is a real number 
Which is mathematical shorthand for
"minus infinity is less than any real number,
and infinity is greater than any real number"
Here are some more properties:
Special Properties of Infinity 

∞ + ∞ = ∞ 
∞ + ∞ = ∞ 
∞ × ∞ = ∞ 
∞ × ∞ = ∞ 
∞ × ∞ = ∞ 
x + ∞ = ∞ 
x + (∞) = ∞ 
x  ∞ = ∞ 
x  (∞) = ∞ 
For x>0 : 
x × ∞ = ∞ 
x × (∞) = ∞ 
For x<0 : 
x × ∞ = ∞ 
x × (∞) = ∞ 
Undefined Operations
All of these are "undefined":
"Undefined" Operations 

0 × ∞ 
0 × ∞ 
∞ + ∞ 
∞  ∞ 
∞ / ∞ 
∞^{0} 
1^{∞} 
Example: Isn't ∞ / ∞ equal to 1?
No, because we really don't know how big infinity is, so we can't say that two infinities are the same. For example ∞ + ∞ = ∞, so
∞  =  ∞ + ∞  which means that: 
1  =  2  
∞  ∞  1  1 
And that doesn't make sense!
I could have also made 1=3 and so on ... so we say that ∞ / ∞ is undefined.
Infinite Sets
If you continue to study this subject you will find discussions about infinite sets, and the idea of different sizes of infinity.
That subject has special names like Alephnull (how many Natural Numbers), Alephone and so on, which are used to measure the sizes of sets.
For example, there are infinitely many whole numbers {0,1,2,3,4,...},
But there are more real numbers (such as 12.308 or 1.1111115) because there are infinitely many possible variations after the decimal place as well.
But that is an advanced topic, and goes beyond the simple concept of infinity we discuss here.
Conclusion
Infinity is a simple idea: "endless". Most things we know have an end, but infinity does not.