the critical strip and critical line
Once you know the zeta function on the whole plane, you can ask where it vanishes. The answer splits cleanly. In the half-plane Re s > 1 the Euler product shows zeta has NO zeros at all. The functional equation then forces zeros in Re s < 0 to sit exactly at the negative even integers s = -2, -4, -6, ... — these are the 'trivial zeros', well understood and uninteresting. Everything mysterious is squeezed into the vertical band 0 <= Re s <= 1, called the critical strip.
Inside the critical strip live all the 'non-trivial' zeros, the ones that actually control the primes. The functional equation's s <-> 1 - s symmetry means these zeros come in pairs reflected across the central vertical line Re s = 1/2, and conjugate symmetry pairs them across the real axis too. That central line Re s = 1/2 is the critical line. It is special because it is the fixed axis of the s-to-(1-s) reflection: a zero on it is its own mirror image. By the prime number theorem (equivalently, zeta does not vanish on Re s = 1), the strip can be narrowed to the open band 0 < Re s < 1, but no further by elementary means.
The critical strip is where the action is because the non-trivial zeros' real parts measure the error term in the prime number theorem: the closer the zeros hug the line Re s = 1/2, the more regularly the primes are distributed. We know infinitely many zeros lie exactly on the critical line (Hardy's theorem), and a positive proportion do, and the first several billion computed zeros all do. The conjecture that ALL of them do is the Riemann hypothesis. A caution: 'critical strip' (a band of width 1) and 'critical line' (a single line in its middle) are different objects — confusing them muddles what is proved versus what is conjectured.
The first non-trivial zero sits at s = 1/2 + i * 14.1347..., right on the critical line. Its mirror partner under conjugation is s = 1/2 - i * 14.1347..., and under the functional equation s <-> 1 - s it maps to itself because its real part is exactly 1/2. The next zeros are at imaginary parts 21.022..., 25.011..., 30.425..., all (so far) with real part 1/2.
Trivial zeros at -2, -4, -6, ...; all non-trivial zeros lie in the strip 0 < Re s < 1, and (conjecturally) on the line Re s = 1/2.
It is proven that the non-trivial zeros lie strictly inside the open strip 0 < Re s < 1 and that infinitely many lie ON the critical line — but NOT that all of them do. The claim that every non-trivial zero has real part exactly 1/2 is the (still open) Riemann hypothesis, not a theorem.