Based on an earlier board game designed to educate the public regarding tax theory, Monopoly, as we know it, was designed and produced by Charles Darrow in 1934 and then sold to Parker Brothers only a year later. Regular tweaks to the look of the game over its long history mean it's possible to date any given Monopoly set with a degree of accuracy.
Look first for the maker. If you have a Monopoly set that doesn't bear a Parker Brothers logo, then it could be one of the earliest sets produced in small numbers by Darrow.
Check the board to see if it is marked with prices. These were added in 1935. Look for the community chest and diamond ring, revisions made in 1949. The absence of either of these point to an earlier model.
See whether the game has any tokens. The Darrow games came without them. The first Parker Brothers set had six (iron, cannon, thimble, ship, shoe, top hat) After the war, new dies were made and a few new tokens were added: the racing car, Scottie dog and horse and rider. In the 1960s and '70s, there were only eight tokens, with the cannon and iron dropped.
Examine the signage on the boxes. In 1936, the words "A Parker Trading Game" were added. In 1951, it was changed to "Parker Brothers." In 1949, "Reg. US Patent Office" became "Registered in US Patent Office." Some of the very earliest Darrow sets came in a large white box, then the packaging went through a number of variations in shapes, size and colour, until the return of the white box in 1962.
Some of the last Darrow sets were issued with a Parker Brothers label.
If buying a vintage Monopoly set, beware so-called "marriages"--a complete set assembled from the pieces of several incomplete ones.