Why do we light vigil lamps in front of icons?

This question was also addressed to Saint Nicholas Velimirovich (1880-1956) in his missionary letters, and this was his answer:

First of all, because our faith is light. Christ said: I am the Light of the world (John 8:12). The light of the candle reminds us of the light with which Christ illuminates our souls.

Second, to remind us of the luminous character of the saint before whom we light the candle – because the saints are called sons of Light (Eph. 5:8).

Third, to serve as a rebuke to us for our dark deeds and for our evil thoughts and desires, and to call us to the path of evangelical holiness, so that we may labor diligently to fulfill the Savior's commandment: so let your light shine before men, that they may see your good works (Mt 5:16).

Fourth, to bring a small sacrifice to God, Who sacrificed everything for us. A small sign of our great gratitude and our luminous love for Him from Whom we ask in prayer for life and health and salvation and everything that only boundless heavenly love can give us.

Fifth, to be a scarecrow for the evil powers, who sometimes attack us even during prayer and divert our thoughts from our Maker - because the evil powers love darkness and flee from any light, especially that which is dedicated to God and His saints.

Sixth, to encourage us to self-immolation. Just as oil and wick burn in a candle, subject to our will, so may our souls burn with the flame of love in all our sufferings, always subject to the will of God.

Seventh, to teach us that just as the candle cannot light without our hand, neither can our heart, this inner candle of ours, light without the holy fire of Divine grace, even if it were filled with all the virtues – because all our virtues are like fuel, and the fire that lights them comes from God.

Source: Answers to Questions of Today's World, vol. I, Sophia Publishing House, 2002