For a standalone application you will need to use Air, FLEX is a framework so you can use it into your Air application.
Air come with SQLLite ready to use, if you dont want to use SQLLite you will have to find a way to communicate with your database : a gateway server (php, java, ...) or found a ready to use socket driver, or write it yourself.
Edit:
Here a starter page where you can found for example Air sample and tutorial
In this one, an example of using an SQLite database. You can download the code of the sample application
What is Flex
Flex is a powerful, open source application framework that allows you to easily build mobile applications for iOS, Android™, and BlackBerry Tablet OS devices, as well as traditional applications for browser and desktop application.
Flex has different SDK Version, more details
The Flex SDK contains hundreds of out-of-the-box components. DataGrids, Charts, Formatters, Validators, and numerous other UI controls are the building blocks for applications of all sizes. Components can be styled and skinned to fit the look and feel you want. There are also hundreds of third party open source and commercial components available for Flex. Mapping APIs, Data Visualization libraries, and Cloud APIs provide the building blocks for assembling great applications.
You will use the following two languages two develop flex applications.
1 . MXML is an XML-based markup language that is primarily used to layout application display elements.
2 . ActionScript is an ECMAScript-compliant object-oriented programming language that is primarily used for application logic
What is Flash Builder
Flash Builder is an enterprise-class Eclipse-based IDE to develop Flex applications.
Flash Builder includes state-of-the-art productivity features (code assist, refactoring, etc),
on-device debugging, and mobile simulators that allow developers to test their applications
on different screen sizes and densities. Flash Builder allows developers to package native
application files (ipa, apk, bar) for release on the Apple App Store, Android Marketplace,
and BlackBerry App world.
What is AIR
Adobe AIR is a cross-operating system runtime that enables you to use your existing ActionScript or HTML/JavaScript development skills and tools to build and deploy rich Internet applications (RIAs) outside the browser(Desktop) and on mobile devices.
Best Answer
I believe @joshtnjala to be partially correct, but I wanted to add on it. If using the Flex 3.3 SDK, you no longer require the AIR SDK as those tools are available to the Flex 3.3 SDK. It is important to note, because I see a lot of confusion or misuse of the term, Flex is actually three things.
Flex SDK (currently 3.3)
allows developers without the Flash and/or Flex IDE's, to develop and compile/publish Actionscript3 and MXML code to flash content (SWF). For example, one could use FlashDevelop for an IDE, and compile the code using the Flex 3.3 SDK.
Flex Framework
(From adobe [http://www.adobe.com/products/flex/overview/][1]) Flex is a free, open source framework for building highly interactive, expressive web applications that deploy consistently on all major browsers, desktops, and operating systems. It provides a modern, standards-based language and programming model that supports common design patterns. MXML, a declarative XML-based language, is used to describe UI layout and behaviors, and ActionScript™ 3, a powerful object-oriented programming language, is used to create client logic. Flex also includes a rich component library with more than 100 proven, extensible UI components for creating rich Internet applications (RIAs), as well as an interactive Flex application debugger.
Flex Builder 3
Flex Builder is merely an IDE to assist in the development of flash (SWF), or AIR content. This builder utilizes the SDK above to compile. The framework may or may not be used depending on the developers needs. Many developers have taken themselves out of the FLASH IDE (THAT NASTY TIMELINE!) and began scripting their AS3 purely in the Flex Builder IDE as it provides many useful features for the "advanced" programmer; Not to mention its a very clean UI and is built on eclipse so it has many plugins and a solid community behind its architecture.
That being said, to expand on what joshtnala said, the difference between using the AIR SKD and the Flex 3.3 SDK, is that if you merely want to build an AIR app, composed of Actiosncript3 code, the AIR SDK is all you need (33mb unpacked). If you intend to use MXML and any part of the "Flex Framework" then you must use the Flex 3.3 SDK (190mb unpacked). This will allow you to use the many available components such as Datagrids, Accordions, etc, in your AIR application. So in the end, both will develop an AIR application as Flex 3.3 SDK incorporates the AIR 1.5 development tools. Abstract your project out, figure out what you will need, if a component works for that, or if a custom implementation is needed, etc, then choose your toolkit.
Hope this helps.