Traditional Christian teaching holds that God is truly one and yet at the same time exists in three "persons," the Father, Son, and Holy Spirit. (See below for details.)
The commandment against graven images in Exodus 20:3-5 (cached) was not applied to the Son because, according to the reasoning of the Second Council of Nicea in 787, he had become man and thus shared in our human ability to be seen and visualized. However, direct depictions of the Father or the Holy Spirit have always been avoided in the East and did not enter western art until the second millenium.
In the first millenium, and still today in the East, the usual way to represent the Trinity symbolically was by depicting "The Hospitality of Abraham" -- God's visit to Abraham at Mamre in Genesis 18:1-15 (cached). The text says it was "the LORD" who visited at Mamre, but Abraham saw three men and Christians take this as a revelation of the Trinity.
In eastern art, Hospitality of Abraham images normally show the three men as angels of very similar appearance sitting at table outside Abraham's tent, attended by him and his wife Sarah (example). One variant type follows Andrei Rubev's 14th century innovation of simplifying the image by removing Abraham and Sarah (example).
In the West, we find examples of the Hospitality of Abraham as early as a sixth century Italian mosaic and as late as Murillo's Abraham and the Three Angels (1667). But in the second millenium the Trinity came to be portrayed directly in the iconographic type known as the "Throne of Mercy," in which the Father sits on his throne and presents his crucified son to the viewer while a dove rests on the cross or hovers just above it (example). This type is still often seen in older Mexican churches (example). Francesco Grandi's bold and joyous Trinity in the Church of San Giacomo in Rome is a 19th century rethinking of the Throne of Mercy.
In the 17th century there was also a brief vogue for representing the Trinity as three identical men (example), conceivably influenced by Hospitality of Abraham images. This period coincided with the Spanish ascendancy in Latin America and the Philipines, so examples can be found in older churches in those areas. One odd example represents the Trinity this way in an image of the Coronation of Mary.
Soon, however, this type of Trinity image was condemned and supplanted by one in which the Father is represented as an older person, the Son as a younger one seated at his right and shouldering a large cross, and the Holy Spirit as a dove that hovers above the space between them. Early examples of this type include a 16th-century sculpture in Burgos and an 18th-century fresco in Ravenna. In the 19th century it became popular in locales as diverse as Navarre and the United States (as at left). An eastern variant omits the large cross but adds Mary.
In the latter type the Spirit is represented as casting light upon the other two persons (symbolically, making it possible for humans to know them), but in one unusual variant he emanates from both their mouths simultaneously, a reference to the Latin trinitarian theology in which the Spirit "proceeds from the Father and the Son" (a phrase from the Roman Catholic text of the Nicene Creed that reads simply "proceeds from the Father" in the Orthodox and Anglican versions).
Feast day (since the 14th century): First Sunday after Pentecost
At left, window in Augusta, Georgia, United States
12/13th century miniature in a vierge ouvrante
A Throne of Mercy painting ca. 1400
15th century Netherlandish painting
Mexican low relief
Trinity altarpiece in the Gesł, Rome
Articles on the theology of the Trinity: