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The Jewish understanding of Satan is very different and more broad than the Christian idea. The Christian concept is inspired by Zoroastrianism and certain Pagan traditions. Zoroastrians believed in two separate gods (Ahura Mazda, god of light, and Angra Mainyu, god of darkness). For Christians Satan is like Angra Mainyu, he is a counter god to "God the Father'. Satan regularly thwarts God's plans etc. This limits God's ability to rule. Christians say for instance that God does not want anyone to perish but that everyone should have eternal life (John 3). But Satan thwarts God's desire and damns the majority of humanity.
For Jews, there is no god nor power other than HaShem and the agents He uses for His purposes.
Jews vary somewhat in how these things are understood, however Satan, as Lucifer, as conceived by the Church, is all but completely rejected by Jews.
Some Jews believe there are beings similar to demons known as dybbuk however most Jews believe these are more the products of Eastern European superstitions (inspired by Christianity during the Diaspora) than actual entities. There is also some consideration of beings "of the desert" similar to the Muslim idea of the Jinn (genies), but again, such minority views are more regionally inspired add-ons from other religions. There are nonetheless a few passages in the Tanakh that does speak of haSatan (The Satan) as an individual being and enemy of the Jews (for instance Zechariah 3:1-2).
"Satan" is not a proper name, it is a descriptor meaning "adversary" and applies to any adversary, as you said. HaSatan (The Satan) can be understood as a proper name however. But in either case, this is understood more as a concept of rebellion against HaShem than as a malevolent entity overseeing it as evil. Again, there are various views and differing opinions, however this is essentially the Jewish understanding. Rambam rejects any notion of a being known as Satan and he carries a lot of weight. Other rabbis do not go as far. In a nutshell, Satan is a metaphor for humans embracing yetzer hara (our natural inclination towards selfishness as taught at Bava Batra 16a).
I'm going to turn this reply into a study when I get the chance. This is an important topic.