This is a problem that is best done geometrically. Remember that |z-a| is the distance from z to a. So means that the distance from z to -i is the same as the distance from z to 1. Therefore z lies on the perpendicular bisector of the line joining -i to 1. The other part of the equation means that z is also on the perpendicular bisector of the line joining 1 to , and that pins down the location of z uniquely.
On the other hand, if you want to solve the equation algebraically, write z=x+iy. Then the equation becomes . Square both sides, and that tells you that . Multiply out the brackets and you'll find that that equation simplifies to the equation of a straight line (which is in fact the perpendicular bisector of the line joining -i to 1, as in the geometric solution).