The Remah (Shulchan Aruch Orach Chaim 576:2) says that some people have the custom to eat eggs at the Seder, as a sign/remembrance of mourning. He posits two reasons:
- The first night of Pesach is always the same day of the week as Tisha Be'av
- To remember the destruction of the Temple. Where it not for the destruction we would be eating the Korban Pesach.
The Mishna Berurah (:11) brings a third reason, from the GR"A:
- As a remembrance for the Chagigah (generic holiday offering) offering that was also brought and eaten that night. [Therefore the egg from the seder plate should be taken and eaten]
We do not make a difference between the first night and the second night - Mishna Berurah S"K 11 and 13 and Be'er Hetiv S"K 2.
No mention of dipping the salt water is made. Askmoses.com mentions that it is a tradition.