Why is the determinant defined in terms of permutations? Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of permutations?
$$
\text{det}(A)=\sum_{p}\sigma(p)a_{1p_1}a_{2p_2}...a_{np_n}.
$$
I have found this one useful:
Thomas Muir, Contributions to the History of Determinants 1900-1920.

I think Paul's answer gets the algebraic nub of the issue. There is a geometric side, which gives some motivation for his answer, because it isn't clear offhand why multilinear alternating functions should be important.
On the real line function of two variables (x,y) given by x-y gives you a notion of length. It really gives you a bit more than length because is a signed notion of length. It cares about the direction of the line from x to y and gives you positive or negative based on that direction. If you swap x and y you get the negative of your previous answer.
In R^n it is useful to have a similar function that is the signed volume of the parallelpiped spanned by n vectors. If you swap two vectors that reverse the orientation of the parellelpiped, so you should get the negative of the previous answer. From a geometric persepective, that is how alternating functions come into play. 
The determinant of a matrix with columns v_1,... v_n calculates the signed volume of the parallelpiped given by the vectors v_1,.. v_n. Such a function is necessarily alternating. It is also necessarily linear in each variable separately, which can also be seen geometrically. 
