The dragon goddess in Fire Emblem: Three Houses is Sothis. She is also known as the progenitor god and is central to the history of Fódlan and the Church of Seiros.
Sothis appears to Byleth as a mysterious girl living within their mind. She initially has no memory of her identity, but she possesses extraordinary power, including the ability to rewind time. The story later reveals that she is an ancient divine being connected to the Nabateans, a dragon-like race that once lived in Fódlan.
Sothis was killed in the ancient past, and Rhea attempted to revive her by placing Sothis's Crest Stone inside Byleth when Byleth was born. That stone eventually allows Sothis to merge with Byleth, giving Byleth the power associated with the Crest of Flames.
A common mix-up is confusing Sothis with Rhea, the archbishop who is also a dragon-like Nabatean, or with other Fire Emblem dragons such as Naga and Tiki. Sothis is the divine progenitor figure specific to Three Houses.