History and precursors in approximate chronological order:
Ancient "strophium" etc.Edit
Often a plain strip of cloth tied or belted just under the breasts, over a simple flowing garment or undergarment
Regency/Empire "short stays"Edit
Not really bras in the modern sense, since the breasts were supported by being pushed up from below
Late 19th century / Edwardian / 1920'sEdit
There were various experimental attempts to get away from the corset before WW1, some more bra-like than others. Note that illustrations from patent applications do not give a good idea of what was typically and commonly worn by contemporary women; illustrations from catalogues, fashion plates, advertisements etc., are better in this respect.
Late 19th centuryEdit
1910's - 1920'sEdit
By the 1920s, the bra existed in a more or less modern form, but it did not immediately catch on.
It probably wasn't really until roughly 1930 that the bra became part of the most usual daily wear of the great majority of ordinary women (during most of the 1920s, combination garments were still often worn).