Login to Joomla Users Table Outside Joomla
You may want to use the Joomla Framework in combination with standalone PHP applications. There can be many reasons for this; such as bypassing Joomla extensions and modules to use different software. Also, in many cases, it is much faster for a programmer to code pure php / mysql than it is to write pure php / mysql and convert it to a Joomla extension. If this is the case, you will probably want to tie into the Joomla Users table. If you do this, the simplest method is to create a login form that encrypts the password with md5. Joomla uses md5+salt by default; but shorter md5 strings work.
Now, your new tables with standalone php scripts can be joined and used with any Joomla tables.