Given a homogeneous polynomial $F$ of degree $d$ in $n+1$ variables $X_0,\ldots,X_n$ and a point $[x_0:\cdots:x_n]$ we cannot evaluate $F$ at that point, because it has multiple such representations, but since $F(\lambda x_0,\ldots,\lambda x_n) = \lambda^d F(x_0,\ldots,x_n)$ we can say whether any such representation (and hence all) vanish at that point.

A projective variety over an algebraically closed field $k$ is a subset of some projective space $\mathbb{P}^n_k$ over $k$ which can be described as the common vanishing locus of finitely many homogeneous polynomials with coefficients in $k$ and which is not the union of two such smaller loci. Also, a quasi-projective variety is an open subset of a projective variety.