I really dislike the expression "the square root of minus one".
Like all non-zero numbers, -1 has two square roots, which we call i and -i.
If x is a Real number then x^2 >= 0, so we need to look beyond the Real numbers to find a square root of -1.
Complex numbers can be thought of as an extension of Real numbers from a line to a plane. The unit in the x direction is the number 1. The unit in the y (imaginary) direction is the number i. So i is called the imaginary unit.
i has the property that i^2 = -1.
If a >= 0 then sqrt(a) means the non-negative square root of a, which lies on the part of the Real line at and to the right of the origin 0.
If a < 0 then we define sqrt(a) to be the principal square root of a, lying on the positive part of the imaginary (y) axis.
So sqrt(-1) = i and -sqrt(-1) = -i.
This all looks like it is working well, but some things break down for square roots of negative numbers. For example, the identity sqrt(ab) = sqrt(a)sqrt(b) does not hold in general:
1 = sqrt(1) = sqrt(-1 * -1) != sqrt(-1) * sqrt(-1) = -1
