How Quality Assurance is related to Testing? Very often I hear people say "Quality Assurance" while talking about "Testing" and vise versa. According to my understanding these two terms mean different things. Could anyone explain the difference? Or they really mean the same?

These two terms are often used interchangeably but this is wrong understanding of their meaning.

Software Quality Assurance is a set of processes/practices which aim to assure high quality of the product. Testing is only one of these processes. QA consists, but is not limited to, design, architecture, coding standards, code reviews, testing, change management, configuration management, release management, audits, etc.

Testing sometimes is the only area where quality assurance is implemented consciously, and this is probably the source of wrong understanding of the terms. If you wanted to use other term for testing it would be quality verification, not quality assurance.

In short: quality assurance is implemented all over the production process while testing is only one of stages of it.