Is there a difference between varieties and affine varieties, if so what is it? I'm reading my Algebraic geometry textbook and wanted to know if they interchange the words varieties and affine varieties.  Just beginning my studies and didn't want to make any assumptions.

It does depend on the definition of variety given in your textbook. Following the conventions of Hartshorne's book Algebraic Geometry (wich is the canonical reference on the subject) he defines over an algebraic closed set $k$ a variety over $k$ as any affine, quasi-affine, projective or quasi-projective variety. Where we say that a variety is


*

*Affine if it is an irreducible closed subset of $\mathbb{A}^n$ with the Zariski topology.

*Quasi-affine if it is an open subset of an affine variety.

*Projective if it is an irreducible subset of $\mathbb{P}^n$ with the Zariski topology.

*Quasi-projective if it is an open set of a projective variety.


So you cannot interchange those terms freely. Every time you start reading an Algebraic Geometry book you should be careful checking what notation and conventions they are following, because they change a lot between authors.
