1.In eclipse i have executed the tests they are working fine, how i have to execute them in production server?
Your tests can run in every environment. You can have a Selenium Hub and many VMs with different operating systems and browsers. But for the beginning the tests can also run from your local computer.
You can choose against what environment (test environment, production) the test run. You can create a config file and set a property for that. Your testing framework can read this file and decide against what kind of environment the tests run. You can also work with Apache Maven, so you can use your tests with Jenkins and configure them.
You can start your test run from your local machine via eclipse or command line, if you choose Apache Maven. You also can have a Jenkins server and start the tests from there with a Jenkins job.
2.selinium Webdriver testing is best testing?
This depends on what exactly you want to test. If you want to test the GUI of web applications in different browsers and on different operating systems, Selenium 2 / WebDriver is an excellent open source tool for that.
3.how to create test suite and how to run them?
Selenium 2 / WebDriver on its own is no testing framework. It just control the browser. The testing framework is JUnit. JUnit provide different ways to create test suites. You can create them with Suites or Categories.
You can also use another testing framework like TestNG. But this are Java testing frameworks only. If you want to develop in another programming language, you have to choose another.
4.What are the steps to test in Chrome browser in Ubuntu 10.0.4 ?
You have to download the Chrome Driver. For some browsers you need special drivers to control them. Not only for Google Chrome. Internet Explorer do also need one. Here is a Getting Started page.
Some other tips: