What is the instance name of the box I'm logged into?
If the DB is running, you can look for the pmon process:

ps auxww | grep pmon | grep -v grep


You'll see one pmon process for each database that is up. For example ora_pmon_YOURDB, where YOURDB is the instance name.

You can also ask the listener which services it is serving connections for, with:

lsnrctl status