What is Web SQL?

Web SQL is simply SQL database that you can run in the web. It is only supported in the latest version of Safari, Chrome and Opera. Web SQL database API  are used to work with client based databases. It is well suited for web applications.

The core methods are:

openDatabase
transaction
executeSql

openDatabase create a database aobject allows you to open an existing database or allows you to create a new one.

transaction allows you control transaction helping you to commit or rollback.

executeSql allows you to execute a Sql queries.

Firefox and Microsoft Edge does not support Web SQL since it is not going to be a W3C standard. However, Web SQL are wrappers for SQLite.

Which is the best browser to test HTML5 pages?

This site (link) tests browsers for the support they provide for
HTML5. They test  various aspects of HTML5 code to arrive at the rating.   

Please access the site for further details here:

http://html5example.net/static/html5test/

These are some of the browsers I use and I am not sure they are
the latest. But you can see that Google Chrome Version 28.0 support most of the functionalities of HTML 5.