Where does the iPhone save contacts?

An iPhone stores nothing at all to the SIM card. The import feature is there for those bringing their own SIM from another device.

Contatcts are stored in the contacts app on the device. If you've enabled iCloud contacts, then they are synchronized to iCloud as well (note iCloud is purely a sync service as far as contacts go, not an archive nor backup service - changes, including deletions, made on the device are reflected in iCloud and vice versa).

Just an addendum to Michael Black's excellent post. You can chose, in Settings>Contacts>Default Account, the account to which contacts are synced. For example, I have both an iCloud account and a Gmail account on my phone. My contacts are synced to my Gmail account.