The escape character is ', so you would need to replace the quote with two quotes.
For example,
SELECT * FROM PEOPLE WHERE SURNAME='O'Keefe'
becomes
SELECT * FROM PEOPLE WHERE SURNAME='O''Keefe'
That said, it's probably incorrect to do this yourself. Your language may have a function to escape strings for use in SQL, but an even better option is to use parameters. Usually this works as follows.
Your SQL command would be :
SELECT * FROM PEOPLE WHERE SURNAME=?
Then, when you execute it, you pass in "O'Keefe" as a parameter.
Because the SQL is parsed before the parameter value is set, there's no way for the parameter value to alter the structure of the SQL (and it's even a little faster if you want to run the same statement several times with different parameters).
I should also point out that, while your example just causes an error, you open youself up to a lot of other problems by not escaping strings appropriately. See http://en.wikipedia.org/wiki/SQL_injection for a good starting point or the following classic xkcd comic.
