A handicap. The key is this: If the word starts with a vowel sound you use 'an'. An apple... An orange... An umbrella etc... In 'handicap' the 'h' is pronounced and therefore you must use 'a.' There are certain words that begin with 'h' such as 'hour' or 'honour' where the 'h' is silent and thus the word starts with a vowel sound. Consequently we say 'an honour' and 'an hour.' There are also some words where the first letter is a vowel but we use 'a.' For example 'a university', 'a union' etc. This is because in these words the word starts with a 'y' sound so we switch to 'a.'

A handicap, as the 'h' is not silent in this word. :-)