Due to my love for crosswords, today I searched the web for a little history of the phenomena, which took the lead during the Roaring Twenties (I was definitely born in the wrong decade).
Here's the first crossword, published in December, the 21st 1913 on the New York World (a New York's newspaper which existed for 70 years only, the last one being published in 1931), and it was created by the English journalist Arthur Wynne. There were only white squares, as the black ones appeared later.

It is interesting that it does change depending on which country they are produced in!
In North America they're usually squared shaped, symmetrical and the black squares have to be maximum a sixth of the totality of squares. Furthermore, the answers have to be at least 3 letters long.
In the U.K. and in Australia, crosswords mantain these characteristics but are allowed to contain a higher number of black squares., meanwhile in Japan those can't be placed along the borders of the "table", nor they can be contiguous to each other.
In Italy there are almost no rules, except the shape which has to be rectangular. Besides, it is possible to have only two letters long answers, black squares can be contiguous to each other and answers can contain more than just one word, to write one attached to the other.
In France, crosswords contain fewer columns per line.
In a Hebrew crossword, the answers must be written from right to left.