The term was in use in this sense as early as 1600, when Samuel Rowlands in his satire The Letting of Humours Blood in the Head-Vaine, referred to 'a Bow-bell Cockney'. John Minsheu (or Minshew) was the first lexicographer to define the word in this sense, in his Ductor in Linguas (1617). However, the etymologies he gave (from 'cock' and 'neigh', or from Latin incoctus, raw) were just guesses, and the OED later authoritatively explained the term as originating from cock and egg, meaning first a misshapen egg (1362), then a person ignorant of country ways (1521), then the senses mentioned above.
The church of St Mary-le-Bow was destroyed in the Great Fire of London and rebuilt by Christopher Wren. After the bells were destroyed again in 1941 in the Blitz of World War II, and before the bells were replaced in 1961, there was a period when no 'true' Cockneys could be born.
Cockney speakers have a distinctive accent and dialect, and frequently use Cockney rhyming slang.