Historically, the study of Diophantine equations included finding integer or rational solutions to a system of equations. This leads naturally to considering an algebraic set over a field $K$: the set of common solutions to a set of polynomial equations in several variables. If the set of solutions over $\overline K$ is irreducible, in that it cannot be expressed non-trivially as a finite union of smaller algebraic sets, then it is a variety.
A variety has a dimension, which coincides with the intuitive notion of dimension. We say the variety is smooth if every point of the variety (over $\overline K$) has an $n$-dimension tangent space. In the LMFDB, we only treat smooth varieties, and consider them in projective space.
Currently, the LMFDB only has $1$-dimensional varieties, i.e., curves. An important invariant of a curve is its genus, which is a non-negative integer. A curve of genus $0$ with a rational point is always isomorphic to $1$-dimensional projective space. A curve of genus $1$ with a point is an elliptic curve, and the LMFDB has elliptic curves over the rationals and over number fields. It has a selection of genus $2$ curves as well.