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How to build web application which will work fine for touch screen?

What all possible support is there from open source world for building web based application for mobile/ipad touch screen?
I was going through http://jquerymobile.com/.. is it good plugin to develop web based application for mobile/ipad touch screens? Has any one used that..is it easy to customize?
what are the major challenges faced in using this?
Any other good tool/plugin for my development work?

PhoneGap is a framework to wrap your HTML5 app in a WebView, so you are able to deploy it like a native app. PhoneGap supports iOS, Android, Windows Phone, Bada, Symbian, Blackberry and WebOS.
Getting started with PhoneGap
jQuery Mobile is a Javascript framework for UI design of apps for mobile touch devices.
Getting started with jQuery Mobile
It is nice and easy to use for simple apps. jQuery Mobile is also themable and you can customize it with your own CSS.
If you want to deploy your web app as a native app in app stores, PhoneGap is the way to go, but you can also deploy your web app with jQuery Mobile UI on a web server, if you want it to be accessible online.
jQuery Mobile Docs:
PhoneGap is an HTML5 app platform that allows developers to author
native applications with web technologies and get access to APIs and
app stores. Applications are built as normal HTML pages and packaged
up to run as a native application within a UIWebView or WebView (a
chromeless browser, referred to hereafter as a webview).
PhoneGap Wiki:
jQuery Mobile: Touch-Optimized Web Framework for Smartphones & Tablets
A unified user interface system across all popular mobile device
platforms, built on the rock-solid jQuery and jQuery UI foundation.
Its lightweight code is built with progressive enhancement, and has a
flexible, easily themeable design.
Seriously cross-platform & cross-device
jQuery mobile framework takes the "write less, do more"
mantra to the next level: Instead of writing unique apps for each
mobile device or OS, the jQuery mobile framework will allow you to
design a single highly branded and customized web application that
will work on all popular smartphone and tablet platforms. Device
support grid
Touch-optimized layouts & UI widgets
Our aim is to provide tools to build dynamic touch interfaces that
will adapt gracefully to a range of device form factors. The system
will include both layouts (lists, detail panes, overlays) and a rich
set of form controls and UI widgets (toggles, sliders, tabs).

Phonegap is another option. Which works for several platforms

We have used jQueryMobile as well as Sencha touch for developing web applications for touch screens.
Sencha Touch is the leading MVC-based JavaScript framework for building cross-platform mobile web applications. Sencha Touch leverages hardware acceleration techniques to provide high-performance UI components for mobile devices. New developers will appreciate the availability of Sencha Architect, a GUI tool that helps create interfaces as easy as drag and drop. 

Any Javascript framework with just one API that supports both Desktop and Mobile?

Any Javascript framework with just one API that supports both Desktop and Mobile (includes phones and tablets)? If not, why isn't there? What happens if I need to port the website from Desktop experience to Mobile experience, do I have to rewrite the entire website?
What do you think about Dojo or Ext JS?

Use Titanium at http://www.appcelerator.com/ for mobile development

Sencha has ExtJS and Touch. Both are very similar in API and worth taking a look.

If you are looking for HTML5 standard based mobile/desktop application, then look at Sench Touch. It works on mobile as well as desktop. Checkout the examples - http://dev.sencha.com/deploy/touch/examples/ - on your desktop (HTML5 compliant browsers) or on your mobile/iPad and see it in action.

Dojo 1.7 works no issue both on desktop and mobile, the only complain for me is the drag and drop API doesn't support touch for now

Is there an equivilent of jQTouch that covers iPhone, Android, WebOS and Blackberry OS?

I am about to start writing a web app for the latest smartphones using HTML5, CSS3 and JavaScript. I don’t need a framework so much as a library like jQuery.
I looked at jQTouch which is great for iPhone and apparently has support coming for Android too but I am looking to cover Blackberry device Software v4.6 upwards and WebOS 5 as well.
Does anyone have any ideas of alternatives to jQTouch or experience using jQTouch with Android's native browser, Blackberry's browser or WebOS native browser?

Well you probably have to use jQuery. In the beginning of October, the jQuery community will release a jQuery Mobile Version, where you can use all the advantages of jQuery within almost every mobile browser.
Check their site out: http://jquerymobile.com/

Sencha Touch is a possible alternative, but I have no idea how well it works on BB and WebOS. I'm afraid that you won't find something that works well with BB, since they have switched to a Webkit based browser only recently.

I did a project for Android and iPhone using jQTouch. There were times when I had to fiddle with the code to get it working correctly but as a whole I found that almost everything worked on iPhone and Android. Where I started having problems was that the Android browser doesn't have nearly as much support for CSS3/HTML5 (especially with animations) and the app ran much slower on android as the browser doesn't have hardware accelerated graphics. This was a while ago so hopefully it has matured a bit now.

PhoneGap seems to be a reasonable cross-platform framework that supports encapsulating web app type development into an iPhone/iOS app, as well seeming to support some of the other platforms you asked about (but I have tried it on those).
http://www.phonegap.com/

Converting Existing site into Mobile Version

The existing site in cakephp with flash. We need to convert it into mobile version.
Please suggest me is there any possibilities for making mobile version.

Take a look at the following frameworks:
For ui and minor ux options
iwebkit http://www.iwebkit.info
(targets both safari and webkit based
browsers)
iui http://code.google.com/p/iui/
For a more advanced ux
jqtouch http://ww.jqtouch.com/ (targets safari, but looks quite good on webkit)
My personal favorite is jqtouch, and as you right now I'm in the middle of porting a cake app to mobile browsers.

If you want to make a real mobile site, you want to look into a framework for that. There are two big ones:
jQuery Mobile is a site based mobile site, you need to create html in your cakephp views. This is served to the user
Sencha Touch is aimed at more sophisticated sites with a lot of interactions. The site is serves as single page, all necessary data if loaded via json. You mentioned CakePHP, so you might want to use Bancha if you decide to use Sencha Touch.

Which javascript framework should i choose if I'm making one site for both medium Desktop PC and mobile phone?

Which javascript framework is most compatible with mobile phones and desktop both?
javascript disabled user of desktop are very low but mobile phone users are high and i heard not all javascript functions work in mobile phones.

See this question (Available iPhone Web Application JavaScript UI Library/Frameworks) for some ideas. Several answers refer to JQTouch (a jQuery plugin), which implies that jQuery may work for you.

there is no js-framework that works on all mobile phones:
if your target is iphone, jqtouch does offer nice iphone-specific stuff on top of jquery
if you want to reach iphone and android (and possibly some other high-end handsets) you could try xui, which is a jquery-like lightweight js-framework (by one of the phonegap-guys)
if your goal is to work on as many phones as possible, you might want to consider building the core of you site/app in standard html and progressively enhancing it with javascript for those
hope this helps,
frank

PhoneGap is an open source development tool for building fast, easy mobile apps with JavaScript.
PhoneGap
Another option you can go for:
baseJS
As for desktop, you know you can use any jquery, prototype, etc

You may want to start with looking at this to get an idea what is supported on the phones you want to support.
http://www.quirksmode.org/m/table.html
On mobile phones you are going to be most concerned with download size, unless you are only targeting phones with a 3G connection, so you may want to write your own framework that is geared toward what you want, specifically.
You will want to write your phone page to work without javascript, then, if there is javascript, then you may want to have it d/l your new framework, so the basic thing that is downloaded is just a bootstrap javascript file that can d/l the real javascript file that will be used.
Your bootstrap file can look at what is available, and if, for example, onfocus is available on a div element then it is probably a desktop, so you d/l the versions that use jquery and d/l the jquery files themselves.