Testing is the key to successful long-term development and is a critical component of the development process. We strongly recommend that you use a test-driven development process, that is, test development that occurs at the same time as code development.
Testing is key to the success of your application, particularly if your application is to be deployed to customers. If you validate that your application works as expected, that there are no unexpected behaviors, your customers are going to trust you more.
There are two ways of testing an application. One is through the Salesforce user interface, important, but merely testing through the user interface will not catch all of the use cases for your application. The other way is to test for bulk functionality: up to 200 records can be passed through your code if it's invoked using SOAP API or by a Visualforce standard set controller.
An application is seldom finished. You will have additional releases of it, where you change and extend functionality. If you have written comprehensive tests, you can ensure that a regression is not introduced with any new functionality.
Salesforce runs all tests in all organizations that have Apex code to verify that no behavior has been altered as a result of any service upgrades.