European options traditionally expire the Friday prior to the third Saturday of every month.
In general, no corresponding formula exist for American options, but a choice of methods to approximate the price are available (for example Roll-Geske-Whaley, Barone-Adesi and Whaley, Bjerksund and Stensland, binomial options model by Cox-Ross-Rubinstein, Black's approximation and others; there is no consensus on which is preferable).
[1] Obtaining a general formula for American options without assuming constant volatility is one of finance's unsolved problems.
An investor holding an American-style option and seeking optimal value will only exercise it before maturity under certain circumstances.
Owners who wish to realise the full value of their option will mostly prefer to sell it as late as possible, rather than exercise it immediately, which sacrifices the time value.
Although these instruments are far more unusual they can also vary in exercise style (at least theoretically) between European and American: