Yes and Yes.

The database is the physical storage of data on disk.
Persists even when Oracle is not running
If the server shuts down, the database still exists because the files remain on disk.

The instance is the running memory and processes that manage the database.
The instance starts and stops, unlike the database files.

Hence, You can have a database without an instance (just like a hard drive without a computer)

You can have an instance without a database, although its pretty much useless without one.

Typically these are 1-to-1 but in the case of RAC (real application clusters) you can have many instances all talking to one database.