What is the difference between an authentic baseball jersey and a replica jersey?

Authentic jerseys are made exactly like the ones the players wear, including sewn on numbers and names on the back (where applicable). Authentic jerseys are sold in numeric sizes.

Replicas are made of lower quality material and often feature numbers and names that are ironed on or silkscreened. Replicas are sold in small/medium/large/etc sizes instead of specific numbers.

Some other differences:

Authentics have longer shirt tails and are cut slightly thinner (the equivalent of an athletic cut). Replicas are a fuller cut with shorter shirt tails, since fans usually don't wear their jerseys tucked in.

Authentics have the Majestic M logo embroidered on the left sleeve. The replica I have has the full Majestic logo (M and the word Majestic below) ironed on to the right sleeve.

Authentics have an MLB logo on the back of the collar. Replicas do not.

Authentics have no size tag on the inside collar. The size is on the Authentic tag sewn on the lower left of the front shirttail. Replicas have the size on the inside collar, and there's a tag on the lower left of the front shirttail that says "replica."

Generally, replicas are just fine for the average fan.