Votive candles are small molded candles, very popular, and destined to be burnt in little holders adapted to their size and shape. Burnt in ideal conditions, a votive candle will last about 15 hours. In ancient times, in the catholic tradition, votive offerings were made to God and/or one or several saints. These offerings were usually accompanied by a specific request to the divinity (protection against a potential or actual danger, recovery from illness, special favour) or were given as a thank you gift for an exauced prayer.

Several hundreds of years ago, these offerings could be anything, with a predilection for an item representing the person whose wish had been exauced. For example, on his wedding day, king Henry the third of England had a golden statue made to the image of his new queen and had this (royal) offering placed on the altar of St Edward in Westminster.
Nowadays, offerings are far more modest and, most of the time, it's a candle that is set to burn to accompany one's prayer or request.

Votive candles are commonly used for that purpose thanks to their small size, relative longevity and the baility to place them in a holder that protects their flame from drafts, quite common in churches. (source : newadvent.org)