A function defined on square matrices. There are many equivalent definitions of the determinant, most of them useful. It is often useful to treat the determinant as a function of n vectors of dimension n. Here are a few:

Geometrical
The determinant of n vectors is the volume of the parallelopiped they span (this is only useful in n-dimensional euclidian space Rn).
Axiomatic
The determinant is a (in fact, "the") function D of n vectors which is linear is each of them, satisfies D(...,a,...,b,...) = -D(...,b,...,a,...) (for every interchange of 2 arguments) and D(e1,...,en) = 1 (for e1,...,en the standard basis vectors (1,0,...,0), ..., (0,...,0,1)).
Explicit formula
The determinant of a matrix
```   a11 ... a1n
a21 ... a2n
.......
an1 ... ann
```
is the sum over all even permutations p of the integers 1...n of the products of ai,p(i), minus the sum over all odd permutations q of the products of ai,q(i).
Recursive formula
The determinant of the matrix above is the sum
i=1,...,n (-1)i+1 a1,i Di,
where Di is the determinant of the (n-1)×(n-1)-matrix you get by deleting column i and row 1 of the matrix above.