Although I was planning on having a discussion for each archangel and not grouping them, these three are so intertwined, it was near impossible to do so unless I wanted you to be constantly going back and forth.
The first basic thing to cover with these three is who they are. I am sure everyone has heard of Lucifer and Satan, but not as many will know of Samael.
Little seems to be truly known about Lucifer, except the fact the he was known as The Morning Star and started the Great War in Heaven due to Pride. There are two alteration of this story. The first being the God created man and wanted the angels to bow to his creation, but Lucifer refused. The other is that God wanted the angels to bow to Jesus.
In mythology, Lucifer is believed to be the son of Aurora (the Dawn) and the father of Ceyx.
Lucifer was a great archangel in Heaven and believed to be either cherubim or seraphim although more point toward cherubim. He is also Michael's opposite, not God's.
Some view Satan as a beautiful being while others see him as a hoofed and horned beast. It is believed that he was the first angel to sin, although some believe the first sin belongs to Lucifer. That's right. Lucifer and Satan are not the same being, contrary to popular belief. The term "devil" was given to Lucifer due to the fact that the concept of fallen angels wasn't widely held yet.
Another name that is generally involved with Satan is Beelzebub. This is also false. Beelzebub was never of angelic descent.
It also believed that Satan was originally a seraphim or cherubim although some believe that he actually never fell nor is an actual evil being but an abstract that was created on the 6th day to test humans, but God still had limits on what he was able to do.
It also strongly shows that Satan was the one that tempted Eve to bite into the the forbidden fruit. Whether he was the snake or just used the snake is up for interpretation, but it is believed he was driven to tempt her due to his jealousy of God's love for Adam. After this, Satan was banished to Earth, which in theory of the Heavens (Sheol aka hell is in 3rd heaven and fallen angel are kept in the second heaven), would mean he is not fallen.
It is also believed he seduced Eve and impregnated her with Cain. Other lovers it is said he has taken have been Lilith (after she left Adam) and three angels of prostitution, Eisheth Zenunium Naamah, and Agrat Bat Mablat.
Samael is believed to have been the angelic name of Satan before his "fall". Samael also translates to "poison of God." It is also believed that Samael was or is one of the angels of death and lives/lived in Araboth, or the seventh heaven.