Assuming that you are not a mathematician, this is what I tell my engineers: What polynomials represent in the mathematical sense are "extension fields" of Galois Fields, which are modulo(2) fields of binary. The smallest prime field that exists is $GF(2)$, and then we represent the others as extension fields of $GF(2^m)$, where $m$ is the number of bits. Again, this is because these are binary fields. These powers represent bit values in the field.
From the practical sense, what is a reducible polynomial? Here is an example:
$x^4+ x^3+ x^1+ 1 =(x^2+ x^1+ 1)(x^2+ 1)$
This is reducible and thereby cannot be used to create a prime extension field. This is why we use irreducible polynomials. The polynomial for AES is
$P(x)=x^8+x^4+ x^3+ x^1+ 1$
Even though this field is not reducible, it still has that $+1$ on it. The short description for the 1 is from set theory, where all elements in a field must form a multiplicative group, and there must me the identity element. Primes can still be multiplied by 1 to get a prime number. The $+1$ is there for completeness and drops out if you do any binary addition between two polynomial values because $+1$ XOR $+1$ is $0$