Some understand that they were Divine names, written and inserted into the pouch formed by the folding over of the rectangular breastplate (choshen) into a square (Rashi, Ramban and cf. Rashbam).
Others interpret them as a reference to the stones of the choshen (or ephod – Otzar Ha’Geonim Berachos responsa p. 4, “Chazal” according to Ralbag, Rambam according to Mirkeves Ha’Mishneh and Ha’Kesav Ve’Hakabalah).
R. Yosef Bechor Shor explains that they were records of the tribal borders [within Eretz Yisrael].
The Talmud (Yoma 21b) states that the Urim and Thummim were “missing” during the era of the Second Temple. Tosafos and Rambam (Beis Ha’Bechirah 4:1 and Klei Ha’Mikdash 10:10) explain that they could not have actually been absent, since the complete set of eight priestly vestments is an absolute requirement for the High Priest, and the Talmud merely means that their prophetic function was not operational. Raavad disagrees, asserting that they were indeed absent, but that there was nevertheless no problem of missing vestments, since “they are not among the enumerated vestments”. The commentators explain that Tosafos and Rambam understand that the Urim and Thummim (at least in this context), refer to the choshen itself (or its stones, as above), which certainly could not have been absent, as it is one of the eight vestments (or at least an integral component thereof), whereas Raavad understands that they were separate entities placed inside the choshen.
My weekly parashah lectures, on the basic question what, exactly, the אורים ותומים were, are available at the Internet Archive.