Just activate the developer mode which can be found in "About odoo" under user at right top. Then db name will be displayed like User(Db name) at right top.

Once you've logged into a database, the URL will contain db=XXXX where XXXX is your current database.

if you need it from inside the code... then
db_name = self.pool.db.dbname (in v7)

or from browser look in the link like ivan explained...
 

To use the external API, you need to parse the instance name, not the database name. If you're running your database on Odoo.sh, you can open the shell to view your instance name.

In code, for version 10, its self.pool.db_name