The Observatory acts as your base, where you can buy, sell, upgrade, create and repair items/weapons (all by talking to Mavda), or regain health by sleeping. You can talk to Elena and give her gifts, which will raise the affinity level between the two. This determines the ending. If you find an unreadable text then you can ask Elena to translate it.
The majority of the game, however, is spent in the Towers. You go through a tower at a time, using the Oraclos Chain and your other weapon to defeat monsters (see more at Battle) and climb up to the top of the Tower, where you will find a Master, or boss. Upon beating a Master, the next Tower will be unlocked.
There is a time limit to how long you can spend in the Towers. The longer you spend in the Towers, the more Elena transforms into a beast. You can temporarily change her back to human by feeding her flesh which you obtain from beasts, but the only way to completely remove the curse is for her to eat the 12 Master flesh, obtained by defeating the Masters. This time limit does not decrease when you're in the Observatory.