A permutation is when you've swapped two indices. So, if I start with the ordering (1,2,3), and I perform one swap (a swap has to be on adjacent numbers), I could get, for example, (2,1,3). Since I've performed one swap, the permutation (2,1,3) is an odd permutation. Now, let's suppose I do another swap from (2,1,3) to get to (2,3,1). I've now done two swaps, so (2,3,1) is an even permutation. The even-ness or odd-ness of a permutation is identified with the even-ness or odd-ness of the number of swaps I must do to get to that permutation. It's a theorem of abstract algebra that, although there are non-unique ways to get to a particular permutation (for example, I might do two swaps and get right back to where I started), the even-ness or odd-ness of a permutation is constant.
Does that help? Incidentally, in practice, the most important thing to know about the Levi-Civita symbol is that
And you should know the relationship between the vector cross product and the Levi-Civita symbol as well.