What is a bitcoin share?

Hmm … you probably are referring to the word share as it is used in pooled mining as part of the method for rewarding miners.

There are dozens of mining pools, and each can have different methods for compensating the miners who help to solve blocks.

Generally, when the word share appears, it refers to the Pay Per Share (PPS) compensation method.  With PPS, work towards solving a block is divvied up into many, many shares.  

Miners then perform work and submit their completed shares back to the pool.  A miner’s compensation when PPS is used directly compares to the number of shares completed and submitted back to the pool.