Affine variety over a field Suppose we have an algebraically closed field $K$. An affine variety is the common zero locus of a collection of polynomials $f_{\alpha} \in K[z_1, \dots, z_n]$. So basically it is the set of points $x$ for which $f(x) =0$? What if $f(x) = 0$ for only some polynomials? What if there is no zero locus? 
 A: When there is no zero, then there is no zero. The easiest example is $1=0$. The empty algebraic set is OK. But it is common to require irreducible spaces to be non-empty, and affine varieties are (often defined as) irreducible algebraic sets and therefore non-empty.
A: In order for a point to be in the variety defined by polynomial equations $f_{\alpha}$, it must satisfy each of the polynomials, not just some.
Also, it is possible for a set of equations to have empty zero locus.  Consider for example the polynomials $x$ and $x-1$ in $K[x]$.  Even though the point $0$ satisfies the first polynomial, it does not satisfy the second, so $0$ is not in the common zero locus of $x$ and $x-1$ (and similarly for the point $1$).  Thus, the common zero locus of these two polynomials will be empty in $\mathbb{A}^1$.
In order that the zero loci of sets of polynomials form the closed sets of a topology, we must have some zero locus be empty.  This topology, called the Zariski topology, is foundational in algebraic geometry.
A: 
Suppose we have an algebraically closed field K. An affine variety is the common zero locus of a collection of polynomials $f_α ∈ K[z_1,…,z_n]$. So basically it is the set of points x for which f(x)=0? 

If the field is algebraically closed, the variety and the set of points it cuts out are almost the same thing, but not exactly.  Because $f(x)=0$ and $f(x)^n=0$ cut out the same set of points but are different conditions algebraically, there is some subtlety that is addressed by the Nullstellensatz.


What if f(x)=0 for only some polynomials?


This has no geometric meaning. The set of $x$ that you get depends on what set of equations you used to cut out the variety from the ambient space, and the same variety can be defined in many different ways.  For example, $X=Y=0$ defines the same locus as $X-Y = 3X+2Y = 0$, but the set of $x$ for which "$f(x)=0$ for only one of the two defining polynomials" changes according to which definition is used.
