The genealogy begins with Jacob's sons and provides information regarding their sons, grandsons, etc., as is necessary to get to both Moses and Aaron. Reuben and his sons are mentioned first, followed by Simeon and his sons, and then Levi and his sons. The genealogies here do not go past Levi's family since Moses and Aaron descended from Levi and are the focus at this juncture.
Levi's three sons are named: Gershon, Kohath, and Merari. Then the names of their sons are also listed. We learn that Moses and Aaron descended specifically from Kohath's son, Amram (who married his aunt Jochebed). Though a marriage between close relatives would later be prohibited by God (cf. Lev. 18), it was allowed at the time. Information is then given regarding Aaron's sons, who would serve as priests after the exodus.
Perhaps the most intriguing thing about this context is the chronological data and how it is interpreted by some scholars. A simple reading of the text would lead to the conclusion that Moses was a son of Amram and a grandson of Kohath. However, other verses suggest that there may be a genealogical omission. Numbers 3:27,28, in particular, states that there were over 8000 men and boys of Kohath shortly after the exodus, and some find that high number to be mathematically impossible in just a couple generations (I'm not as troubled as some by this because of Exo. 1:7, but I can still see the difficulty). Anyway, the solution suggested by some is that there are two Amrams in the lineage (though only one is mentioned), with an unknown number of generations in between. This is certainly possible since we have a record of something similar happening in Ezra 7:3 and I Chronicles 6:7-11.
I wonder also if there is a desire to insert more generations into the text in order to help fill the span of time the Israelites were enslaved in Egypt. What I mean is this: It is mathematically challenging to compute 400 or more years of Israelite history in Egypt if Exodus 6 contains a complete genealogy. We will elaborate more on this when we come to Exodus 12:40,41, which is a fascinating study in chronology when compared with Galatians 3:17.