There are a few ways of integrating with QuickBooks, but if it's a web application, your best bet is to use the QuickBooks Web Connector, or to use a commercial package for SQL access.
The Web Connector is the only Intuit-supported solution to integrating with QuickBooks from a web application. Basically, you build a SOAP server / Web Service which speaks a specific set of methods. The Web Connector then is installed on the machine running QuickBooks, and polls your web service asking "Hey, got anything for me to do?" Your web service can then respond with qbXML requests (examples of qbXML here: QuickBooks qbXML Examples) which tell the Web Connector "Add this customer: ..." or "Send me invoices which match: ..." or etc. etc. etc. The Web Connector then relays those requests to QuickBooks, QuickBooks processes them, and the response is relayed back to your web service. Your web service might then process the response somehow, and then send the next request over to the Web Connector. This can continue in a loop for as long as you like, continuing to send requests to QuickBooks and receiving the responses.
You might also look into AccessBooks and QODBC. Both are commercial wrappers around the QuickBooks qbXML SDK which allow direct SQL access to QuickBooks.
Here is some additional information on my QuickBooks Integration Wiki about how to integrate applications with QuickBooks.
You might also want to search around on Google. I know there were some QuickBooks Rails packages/gems out there somewhere, but last I looked it did not support the QuickBooks Web Connector. Instead, the Rails application had to be installed on the same machine as QuickBooks, and it communicated via COM.