An indefinite article is a grammatical article used to express indefiniteness (i.e. lack of definiteness). In English, there are two of these; 'a' and 'an'. They provide a grammatical marker that the following noun is both (1) countable and (2) singular.
'A' is generally used before consonant sounds while 'an' is used before vowel sounds. Note that this reflects speech sounds rather than spelling. So, for example we say "an hour" not *"a hour" and "a university" not "*an university".
Low level learners will often say "a" when they need to say "an" - this is usually worth correcting. Intermediate and advanced speakers sometimes slip and use 'a' before a vowel sound, particularly if there is a pause between the article and noun; this includes native speakers, so it's probably not worth correcting if it's a slip. Correct in writing.
Chinese-speaking students will often drop articles because Chinese doesn't have them.
Contrast the definite article.