Examples of a binomial are
Examples 
Explanation of example 
x + 3 
This expression has two terms, 'x' and '3' that are not like.

x^{2} + 5 
This expression has two terms, 'x^{2}' and '3' that are not like.

x^{2} + x 
This expression has two terms, 'x^{2}' and x' that are not like.

x + 3 +2 
This expression actually can be simplified to x + 5 which is an expression that has two unlike terms. 
x + x + 3 
This can be rewritten as 2x +3 which is an expression with two unlike terms.

On the other hand, x+2x is not a binomial because x and 2x are like terms and can be reduced to 3x which is only one term. Remember, a binomial needs to be two separate terms that cannot be combined further.
Non examples of a binomial are
Examples 
Explanation of example 
3 + 2 
This expression can be rewritten as 5, which is only a single term. Remember we need 2 unlike terms for a binomial

x^{2} 
This expression only has 1 term. 
x + x 
This expression can be rewritten as 2x, which is only a single term. Remember we need 2 unlike terms for a binomial

x^{2} + 3x + 5 
This expression has three terms. (Not a binomial but actually a trinomial) 