An altar-candlestick consists of five different parts; the base, the stem, the knob in the middle of the stem, the bowl to collect the wax drippings, and a tube to hold the candle. In the early days of the church candles weren't allowed on the altar. Instead they used the light from candles that were hung in chandeliers above or next to the altar. The acolytes (candle bearers) would carry the candles in procession to the altar then set them next to the altar on the floor of the sanctuary, as is still the custom in the Eastern Church. It wasn't until the tenth century that candles were permitted to be placed upon the altar because beforehand people felt that only the relics of saints and the book of Gospels should be placed upon the altar.
The custom of placing two candles on the altar became general in the sixteenth century. These two candles are meant to represent "the joy of two Peoples who rejoiced at the birth of Christ," (i.e. Mary and Joseph). This is also where the Marian shrine, or altar on the Gospel side of the sanctuary and the shrine or altar dedicated to Joseph on the Epistle side also came from.
Only two candles were used then because Mass were being said as Low Mass. But on feast days and holy days four or six candles were used. So with the spread of the High Mass came the use of six candles set upon the altar.
According to the "Caeremoniale Episcoporum" (I, xii, 11), there should be on the high altar six candlesticks and candles of various sizes, the highest of which should be near the cross. If all six be of the same size they may be placed on different elevations, so as to produce the same effect; a custom, however, has been introduced of having them at the same height and this is now permissible (Cong. Sac. Rit. 21 July, 1855).
On the other altars of the church there should be at least two candlesticks, but usually four are used; on the altar of the Blessed Sacrament, if the Blessed Sacrament is not kept on the high altar, there should regularly be six.
The Roman Missal (Rubr. 20) says also that a third candle should be placed on the epistle side, and that this extra candle should be lit at low Masses from the consecration to the consumption of the Precious Blood. The third light is not usually placed on the altar itself, but on the credence, or on the step of the altar at the place where the altar-boy kneels. But in roman custom the extra candle may be set on the epistle side of the altar because in most churches in Rome there is no space for credence tables, and the sanctuary's were not designed for them, so the epistle side of the altar is used to set the water and wine, this extra candle, and any other items necessary to say Mass on.
The altar boy would light this candle at the consecration and hold onto it, carrying it next to the priest and other altar boy (who held the paten) as the priest distributed communion. This way the Eucharist always has a candle to signify the real presence of Jesus in it and it represents the faithful, who are represented by the candle as having their thoughts and prayers and adoration before the Blessed Sacrament. The same concept is applied to the votive candle placed besides the tabernacle. It's as though the altar server takes that votive candle next to the Blessed Sacrament wherever it travels, such as communion.
(Which by the way, is what Fr. Grondz has done on both the Mary and Joseph side altars along with the use of unbleached candles for the Lenten season.)