Zero, sight height, and the carbine's geometry
Zero is the distance at which your line of sight and the bullet's path cross so that point of aim equals point of impact. Two things make the carbine's zero behave differently from a precision rifle's: the optic sits high over the bore (roughly 2.5 inches on an AR-pattern rifle, vs. ~1.5 on many bolt guns), and the bullet is light and fast with a flat-ish but quickly-dropping arc.
Because the sight is well above the bore, the bullet starts about 2.5 inches below your line of sight and has to rise to meet it. It crosses your line of sight once on the way up (the near zero), continues above the line of sight through its mid-range, then falls back through the line of sight again (the far zero). Understanding this two-crossing geometry is the whole key to carbine holds inside 300.
Why 50/200
The 50/200 zero (also called the "improved battlesight zero") sets the near zero at 50 yards; the geometry of a typical 5.56 load then puts the far zero at roughly 200 yards. Between and around those crossings, the bullet never rises more than ~2 inches above or falls more than a couple inches below your line of sight all the way out to ~250 yards.
That's the magic: with a 50/200 zero you can hold dead-on, center mass, from the muzzle to about 250 yards and never be off by more than a couple inches — no holdover thinking required in the distances where most carbine work happens. It's forgiving, fast, and simple. This is why it's the dominant fighting-carbine zero.
The main thing to know: inside 50 yards the bullet is still below your line of sight by up to ~2.5 inches (it hasn't risen to meet your aim yet), so at very close range you actually hold slightly high — see the offset lesson.
The alternatives, briefly
A 100-yard zero is simple and precise at 100 but requires holdover sooner past that and gives a larger mid-range under-the-line dip up close. A 36/300-type zero maximizes point-blank range for a torso target but puts the bullet several inches high at mid-range, which is too much for precise work. For most users on most carbines, 50/200 is the best default; choose another only with a specific reason. Whatever you pick, know the resulting trajectory — the zero is only useful if you know where the bullet is at every distance relative to your aim.
Confirming the zero
Zero at the actual near-zero distance if you can (50 yards), then confirm at distance (200 yards) rather than trusting the theory — your specific load, barrel, and optic height will shift the far zero a little. Fire a group, adjust to center it, and verify. As always: zero from a stable, supported position so you're correcting the rifle, not your wobble, and confirm with enough rounds (5) that you're adjusting off a real group center, not one lucky shot. Record the zero and the resulting holds in your DOPE just like a precision rifle.