I have made . . . . . . . . mistake.

The indefinite article comprise 2 forms. "a" when it precedes a word starting with a "consonant", & "an" when it precedes a word starting with a "vowel". The indefinite article denotes the noun is a general idea instead of a specific thing.
In the above we used "a", indefinite article, before the word "mistake" since the word starts with a consonant and it does not specify or restrict the meaning of the word.