the principal argument
Since the argument arg z is multivalued — every full turn of 2 pi gives the same point — we often want to pick one specific angle out of that endless family and call it 'the' argument. That chosen representative is the principal argument, written with a capital A: Arg z. The standard convention is to choose the unique angle lying in the half-open interval from -pi (exclusive) to pi (inclusive), so that every nonzero z has exactly one principal argument.
Concretely, Arg z is the angle you would naturally name if you swept counterclockwise no more than half a turn, or clockwise less than half a turn, from the positive real axis. The positive real axis has Arg z = 0; the positive imaginary axis has Arg z = pi/2; the negative real axis has Arg z = pi; and a point just below the negative real axis has Arg z just above -pi. The full argument is then arg z = Arg z + 2 pi k for any integer k.
The point of fixing a principal value is to make the argument an honest single-valued function — but you pay a price. Because the interval has a hard edge at -pi and pi (the negative real axis), Arg z jumps suddenly by 2 pi as you cross that ray: it leaps from near pi to near -pi. That discontinuity is unavoidable, not a flaw of the particular choice, and it is the prototype of a branch cut, which you meet again with the complex logarithm.
Arg(-1) = pi, but Arg(-1 - 0.001 i) is just above -pi: a tiny step below the negative real axis drops the principal argument by almost a full 2 pi. The jump lives along the negative real axis.
Pinning the argument to (-pi, pi] forces a 2 pi jump across the negative real axis — the prototype branch cut.
Conventions differ: some books use [0, 2 pi) instead of (-pi, pi]. Neither is more 'correct'; just state your convention. Whatever you pick, the principal argument is discontinuous along one ray out of the origin.