This tense is considered to be the bugaboo of French grammar (or at least one of them). The descendant of the Latin perfectus, it was once used in a three-tense system which also included the imperfect and the passé composé. The passé simple and the passé composé shared functions which in modern French are performed by the passé composé alone. Formerly, the passé composé could only describe discrete past events which had some kind of connection to the present; the passé simple described events which have no connection to the present. An equivalent in English would be the difference between "I have eaten" (connection to the present) and "I ate" (no connection to the present). Nowadays use of the passé simple is limited to formal written French, such as literary and historical works.
The tense causes problems because of its forms. Here is chanter, for example (with the endings in bold):
je chantai nous chantâmes
tu chantas vous chantâtes
il/elle chanta ils/elles chantèrent
These endings, especially in the plural, are very unusual in French. This is further complicated that many verbs have completely irregular stems in the passé simple which are unrelated to any of their other stems. An example is naître (to be born), of which the stem is naqu-. Thus:
je naquis nous naquîmes
tu naquis vous naquîtes
il/elle naquit ils/elles naquirent
Fortunately most students of French (and most French people themselves) will only ever have to understand this tense, never to use it.