QA involves the entire software development PROCESS - monitoring and improving the process, making sure that any agreed-upon standards and procedures are followed, and ensuring that problems are found and dealt with. It is oriented to 'prevention'.
Testing involves operation of a system or application under controlled conditions and evaluating the results (eg, 'if the user is in interface A of the application while using hardware B, and does C, then D should happen'). The controlled conditions should include both normal and abnormal conditions. Testing should intentionally attempt to make things go wrong to determine if things happen when they shouldn't or things don't happen when they should. It is oriented to 'detection'.
Quality means that a software applicaton, architecture, application technologies, network, system, hardware and configuration are reasonably bug-free, delivered on time and within budget, meet requirements and/or expectations, and are maintainable. However, quality is obviously a subjective term. It will depend on who the 'customer' is and their overall influence in the scheme of things. A wide-angle view of the 'customers' of a software development project might include end-users, customer acceptance testers, customer contract officers, customer management, the development organization's management/accountants/testers/salespeople, future software maintenance engineers, stockholders, magazine columnists, etc. Each type of 'customer' will have their own slant on 'quality' - the accounting department might define quality in terms of profits while an end-user might define quality as user-friendly and bug-free.
