“Participer à” means to “participate in.” this phrase would refer to a noun, like participating in some event.
Example: Je participe à la danse.

“participer au” means to “participate in the”
Example: Je participe au festival.

“au” is “à” + “le.” It is also changed for “les” becoming “aux”

“participer de” means to “participate in” refering to a verb/activity.
Example: Je participe de courir.

The difference in meaning is between participer à (that changes to au / à la / aux according to the following word), and participer de.

Participer à is a synonym of prendre part à (to take part in).

• Des milliers de personnes ont participé à la « marche pour les sciences ».

• J’ai participé au concours de photos.

à le is never said or written, it is always contracted into au.

• Les femmes ne pouvaient pas participer aux Jeux Olympiques dans l’Antiquité.

à les is never said or written, it is always contracted into aux.

Participer de can be replaced by tenir de or appartenir à. It means "to be similar in nature to".

• Le mulet participe de l’âne et du cheval.

which most people would probably phrase as: " Le mulet tient de l’âne et du cheval" since participer de is rarely used in French.

• L’amour participe de l’âme même. Il est de même nature qu’elle. (Victo Hugo, Les Misérables)

Apart from their meaning, perfectly explained by Laure above, you should keep in mind that ‘participer de’ is hopelessly outdated, while ‘participer a/au’ is still in abundant daily use.