There is no standard item positioned behind the eternal light in a synagogue. It depends on the decor of the synagogue, and not on Jewish law.
The "eternal light" in the synagogue is not a permanent yahrtzeit light - it is a sign of the eternal covenant between G-d and the Jews.
the ark, eternal light and other
The ner tamid, or eternal light, is a reminder of the lamp in the Beis Hamikdash that never went out.
A Torah scroll, a Holy Ark in which to place the Torah scroll, and the eternal light. Also: seats and prayerbooks.
Most commonly, a small light that hangs directly over the door to the ark where the Torah scroll is stored.
it's called ner tamid (× ×¨ תמיד), pronounced nehr tah-MEED
A typical synagogue contains an ark (where the scrolls of the Law are kept), an “eternal light” burning before the ark, two candelabra, pews, and a raised platform (bimah), from which scriptural passages are read and from which, often, services are conducted.
The Ner Tamid or Eternal Flame is a representation of the eternal flame that existed in the Great Temple in Jerusalem. In that Temple, there were flames on the sacrificial altar that were always burning. This constant light is a symbol of God's constant presence.
A ner tamid, also known as a sanctuary lamp, is "is a light that shines before the altar of sanctuaries in many denominations of Jewish" worship. "Hanging or standing in front of the ark in every Jewish synagogue, it is meant to represent the menorah of the Temple in Jerusalem as well as the continuously burning fire on the altar of burnt offerings in front of the Temple.[2] It also symbolizes God's eternal presence, and is therefore never extinguished." In the event of a power outage alternate energy sources keep the ner tamid lit.
Perhaps you are asking about the eternal light? Synagogues have an eternal light in front of or above the ark, the ornate cabinet holding the scrolls of the Torah used for public reading during worship services. The light is a symbol of God's presence and a symbol of the eternal light that used to burn in front of the Holy of Holies in the Temple in Jerusalem back before the Roman Empire destroyed that and made off with the lampstand (you can see the image of that booty on the Arch of Titus in Rome).
Every synagogue has a "Ner Tamid" - an Eternal Light somewhere towards the front. Probably reminiscent of the Menorah in the Holy Temple in Jerusalem which had a lamp that burn all day long. Sometimes synagogues have one or more "Ner Neshama" - Soul Lamps. It's customary to light a candle (or electric lamp) in memory of a deceased, and some people pay the synagogue to keep these lamps burning - so they don't have to keep them at home. Some synagogues have a "Wunder Lamp" - "information" lamps which is lit to inform everybody of the presence of people celebrating something; a Mohel, Sandek, Bar Mitzva boy, groom or mother on her first visit to the synagogue after giving birth. (Double click on words to discover their meaning.)
the eternal light is used in the Jewish festival of Hanarashi