n dimensional affine space $A^n$ over $k$ is the set $k^n$. The components of a vector are its coordinates. $A^1$ and $A^2$ are called the affine line and plane respectively.

n dimensional projective space $P^n$ over $k$ is the set of lines through the origin in $k^{n+1}$. $P^1$ and $P^2$ are called the projective line and plane respectively. The coordinates of a vector in $k^{n+1}-\left\{0\right\}$ are called the homogenous coordinates of line spanned by it. There is (noncanonical) decomposition of $P^n=A^n\cup P^{n-1}$, where $P^{n-1}$ corresponds to the set where the last coordinate is $0$. When $n=1$, we can also see this as follows: every line in $k^2$ is determined by it's slope which is either finite (in $A^1$) or $\mathrm{\infty }$.

Given a set $S$ of polynomials (respectively homogeneous polynomials) in $k[x_1,...x_n]$ (respectively $k[x_0,...x_n]$), let $V\left(S\right)$ (respectively $V_P\left(S\right)$) be the set of zeros of these polynomials in $A^n$ (respectively $P^n$). An algebraic subset of n-dimensional affine (respectively projective) space is a set of the form $V\left(S\right)$ ($V_P\left(S\right)$). These are also called closed sets since they are precisely closed sets for the Zariski topology.

When $X$ is a subset of $A^n$, let $I\left(X\right)$ be the set of polynomials in $k[x_1,...x_n]$ which vanish on $X$. $I\left(X\right)$ is always a radical ideal. The Hilbert Nullstellensatz states that $I$ and $V$ are inverse operations yielding a bijection between the collection of algebraic subsets of $A^n$ and radical ideals in $k[x_1,...x_n]$. As a corollary, we have the weak Nullstellensatz $V\left(I\right)$ is empty if and only if $I=\left(1\right)$.

The set of complements of algebraic sets in affine or projective space forms a topology called the Zariski topology. Any subset gets an induced topology which goes by the same name. If k is a topological field such as C, then affine and projective space carries a second topology called the usual topology which is finer than the Zariski topology.

An affine or projective algebraic set is called a variety if it is irreducible in its Zariski topology i.e. if it cannot be written as the union of two proper closed sets. An algebraic subset $X$ of $A^n$ is a variety if and only if $I\left(X\right)$ is prime.

A quasiprojective variety is an open subset of a projective variety. Projective varieties are clearly quasiprojective. Affine varieties are also quasiprojective, $X$ in $A^n$ is an open subset of its closure in $P^n$.

An algebraic map or regular map or morphism of quasiprojective varieties is a map of whose graph is closed. The collection of quasiprojective varieties and morphisms forms a category. An isomorphism of quasiprojective is an isomorphism in this category; more concretely it is a bijection for which it and its inverse are both morphisms.

If $X$ is affine variety in $A^n$, then its coordinate ring $A\left(X\right)=k[x_1,...x_n]/I\left(X\right)$. The functor $X\to A\left(X\right)$ is an antiequivalence of the full subcategory of affine varieties and the category of affine domains over $k$ (finitely generated $k$-algebras which are integral domains). In particular, a morphism of affine varieties $X\to Y$ is an isomorphism if and only if the induced homomorphism $A\left(Y\right)\to A\left(X\right)$ is an isomorphism of algebras.

The category of quasiprojective varieties has products which refines the set theoretic product. The product of $A^n$ and $A^m$ is isomorphic to $A^{n+m}$. The product of two affine varieties is affine and its coordinate ring is the the tensor product of the coordinate rings of the factors.

The product of projective varieties is projective thanks to the Segre embedding $P^n\times P^m\to P^{\mathrm{nm}+n+m}$.
The map sends a pair to the product of homogeneous coordinates of the pair.

An algebraic group is a group $G$ in the category of quasiprojective varieties i.e. $G$ is simulateneously a group and variety and the group multiplication $G\times G\to G$ and inversion $G\to G$ are morphisms.

A homogenous space is a variety $X$ such that there is an algebraic group $G$ and a transitive action on $X$ for which $G\times X\to X$ a morphism.

A nonconstant polynomial $f$ in $k[x_1...x_n]$ defines a hypersurface $V\left(f\right)$ in $A^n$. A point $a$ of $V\left(f\right)$ is called a singular point if all the partial derivatives of $f$ vanish at $a$. The definition can be extended to all varieties and it can be made more intrinsic. In particular an isomorphism takes a singular point to a singular point. When $k=C$, a nonsingular variety (i.e. a variety whose points are all nonsingular) is a complex manifold.

The function field of an affine variety is the quotient field of its coordinate ring. An element of the function field, called a rational function, can be represented by a regular function on a nonempty open set. This leads to a defintion of function field for quasiprojective varieties..

Two varieties are birational if they have isomorphic function fields. Equivalently, two varieties are birational if they have isomorphic open sets. Such and isomorphism is called a birational equivalence.