SQLite has a master table which contains the structure of the DB.

To list just the user generated actual tables (not index'es and excluding the statistic tables that SQLite maintains):

Use a select statement like this (tested on one of my DB's using SQLite Manager):

SELECT name FROM sqlite_master WHERE TYPE = 'table' AND 
    name NOT LIKE 'sqlite_%';
[download]

Prepare this select statement, execute it and fetch the resulting rows in the normal Perl way.

For fun, SELECT * instead of tbl_name and see what all is in this thing. For example includes the SQL code that generated the table, etc.

I like the SQLite Manager browser plugin, but for security reasons, it doesn't run on Chrome or Firefox anymore. So I installed Waterfox 64 bit browser just for the purpose of running this one plugin. I don't use Waterfox for any real browsing.

Update: I don't know if there is a standard SQL way of getting this information or not? That select statement is specific to an SQLite DB. So it won't work on another DB. But this will get you what you want from SQLite.