You enter the Volkihar Undercroft from the secret entrance on the back of the island. Climb down the stairs and open a big wooden door. There will be several Death Hounds in this area, later joined by Feral Vampires. Kill these enemies. Listen to the instructions from Serana who guides you on where to go. Go up some stairs. Activate a lever there. Then go back down. When you reach a certain spot, Serana tells you to take a left. It is one of those weird double barred security measures her father put in when he got more paranoid. If you follow the path around, you can find the other switch.

Go through some flooded tunnels. You reach a portion blocked by spider webs. Use your weapon to cut through the webs. There is a giant Frostbite Spider behind these webs, so prepare to kill it. Behind the giant spider, near the window, there will be another lever. Activate this second lever.

From the narrow bridge, go west this time, and up some long stairs. Serana tells you that this leads out to the courtyard, and to just head from the door. Exit the Undercroft and to the Volkihar Courtyard.

In the center of the courtyard, there is a big Moondial. Serana tells you there is something strange with the moon dial. Investigate the Moondial. This is a PUZZLE

Volkihar Courtyard Moondial Puzzle Solution
If you examine the circular tiles on the ground around the moondial, you will see some of them have missing crests.

Full Moon Crest
Leaning on a wall in the east side of the courtyard.

Skyrim Dawnguard Mooncrest

Half Moon Crest
Submerged in a small pool of water in the north side of the courtyard.

Skyrim Dawnguard Moondial Crest

Crescent Moon Crest
Climb up two flights of stairs to the Northeast and find this crest on the floor near a round wooden table.

Skyrim Dawnguard Moondial Crest

Place each crest in the proper slot, based on the phases of the moons. If you are not placing it in the correct tile, it will say that "you do not have the proper crest", so just try another tile.

Once you have placed all the crests in their position, the moondial spins, and spiral stairs leading down into the ground are revealed.