Q:

What is the meaning of "reduced schemes of finite type"?

What does the following term mean?

category of geometrically reduced schemes of finite type over some field

(I know what a category and a field is but I cannot translate any of the middle bits)

A:

You need to consult a text on scheme theory, for instance
Hartshorne's Algebraic Geometry. A scheme of finite type
over a field $k$ is one with a finite cover of spectra of
rings of the form $k[x_1,\ldots,x_m]/I$ where $I$ is an ideal.
A reduced scheme is a scheme where for each open set $U$ the ring
of functions defined over $U$ has no nonzero nilpotent elements.
It's geometrically reduced if it remains reduced when the base
field is changed to its algebraic closure.

A:

It is probably worth mentioning that another name for this is the category of abstract (not necessarily separated) algebraic $k$-varieties.  

