In the context of finite fields, the definition of a primitive element $\alpha$ is given by: $\alpha$ is primitive if it generates all elements of $F_q - \{0\}$ when raised to powers up to $q-1$.
And for definition of a primitive polynomial: we say that an irreducible polynomial that $\alpha$ satisfies is also primitive.
To me it's unclear what is meant by $\alpha$ satisfying a polynomial, although my guess is that $\alpha$ is a zero of the polynomial. Also, I ask this question because I do not understand the definition of a primitive polynomial so if someone has a different/clearer definition(or explanation) then I would appreciate that.
That's exactly what it means. Ie, $a$ satisfies the polynomial $f(x)$ iff $f(a) = 0$.