The fundamental theorem of algebra says that a non-constant polynomial over an algebraically closed field has a root.
Note that $f(x) \in \mathbb C[x]$ is a non-zero constant if and only if the ideal
$\bigl(f(x)\bigr)$ is the unit ideal. Since any ideal in $\mathbb C[x]$ is principal, another way to phrase the FTA is as follows: if $I \subset \mathbb C[x]$ is a non-unit ideal, then there exists $z \in \mathbb C$ such that $f(z) = 0$ for
all $f(x) \in I$. (The converse also holds, more or less obviously.)
The Nullstellensatz says the same thing for an ideal in $\mathbb C[x_1,\ldots,x_n]$,
namely: an ideal $I$ in this ring is non-unit if and only if there exists $(z_1,\ldots,z_n)$ such that $f(z_1,\ldots,z_n) = 0$ for all $f \in I$.
(Again, the if direction is obvious, but the only if direction is non-trivial
--- although actually, in the case of when the field is $\mathbb C$, it is not
so difficult to prove, because $\mathbb C$ is so much bigger than its prime subfield $\mathbb Q$.)