Polynomials
A polynomial looks like this:

example of a polynomial
this one has 3 terms 
It can be made of:
That can be combined using:
+  × 
addition, subtraction and multiplication, ... 

... but not division! 
Those rules keeps polynomials simple, so they are easy to work with!
Polynomials or Not?
These are polynomials:
 3x
 x  2
 3xyz + 3xy^{2}z  0.1xz  200y + 0.5
And these are not polynomials
 2/(x+2) is not, because dividing is not allowed
 3xy^{2} is not, because the exponent is "2" (exponents can only be 0,1,2,...)
But this is allowed:
 x/2 is allowed, because it is also (½)x (the constant is ½, or 0.5)
 also 3x/8 for the same reason (the constant is 3/8, or 0.375)
Monomial, Binomial, Trinomial
There are special names for polynomials with 1, 2 or 3 terms:
How do you remember the names? Think cycles! 

(There's also quadrinomial (4 terms) and quintinomial (5 terms), but these are not often used)
Lots and Lots of Terms
Polynomials can have as many terms as needed, but not an infinite number of terms.
What is Special About Polynomials?
Because of the strict definition, polynomials are easy to work with.
For example we know that:
So you can do lots of additions and multiplications, and still have a polynomial as the result.
Degree
The degree of a polynomial with only one variable is the largest exponent of that variable.
Example:

The Degree is 3 (the largest exponent of x) 
For more complicated cases, read Degree (of an Expression).
