BetaHedrons are Kryptonian devices that were used as secondary sources of power on Krypton, specifically used to power probes that scientists sent to space. According to Kara Danvers, it is a "smaller" version of the Omegahedron, though it is still powerful enough to provide a near-unlimited power supply.
Krypton's scientists periodically sent probes with knowledge about their planet, society, and religion all across space in order to teach other civilizations about Kryptonian culture. These probes were powered up by BetaHedrons. At some point, one of these probes landed on Earth.
After he was saved by Supergirl, Thomas Coville found the probe while traveling around Earth, and used the knowledge inside to create the Cult of Rao. In order to recruit more believers to the religion, he turned the BetaHedron unstable in the basement of a sports stadium, knowing it will eventually explode, counting on Supergirl to stop it. Unbeknownst to him however, the probe contained a soil sample from Krypton laced with Green Kryptonite, which weakened Supergirl. Fortunately, with the aid of Alex Danvers, Supergirl contained the blast by pushing the probe into a deep hole she burned in the ground. Once the BetaHedron exploded, the resulting tremor impacted the Legion's cruiser, resulting in the awakening of Mon-El.