The Hasmoneans were a Jewish priestly dynasty that ruled Judea from roughly 164–63 BCE.
Origins

They emerged from the Maccabean Revolt, led by Mattathias and his sons — most famously Judah Maccabee — against the Seleucid Greek king Antiochus IV, who had desecrated the Jewish Temple in Jerusalem. The Jewish holiday of Hanukkah commemorates their victory.

Who They Were

They were a priestly family from the tribe of Levi, specifically from a line called the Hasmon — hence the name Hasmonean. They were thoroughly Jewish, not Roman.

At Their Peak

They ruled an independent Jewish kingdom, controlling most of the land of ancient Israel. They served as both kings and high priests, which was actually controversial among some Jews who felt the two roles shouldn’t be combined.

Their Decline

Internal family power struggles weakened them significantly. The Roman general Pompey exploited this in 63 BCE, marching into Jerusalem and bringing Judea under Roman control. The Hasmoneans lingered as puppet rulers for a while before Herod the Great — an Idumean appointed by Rome — replaced them entirely around 37 BCE.

Their Relation to Jesus
By the time Jesus was born, the Hasmonean dynasty was already finished. They were his distant historical backdrop, not direct relatives or associates.