Irrespective of means without being influenced or affected by something, or regardless of it. When you make a decision irrespective of the weather, you're going forward with your plan whether it rains or shines. When a teacher says everyone will take the test irrespective of their excuses, she means the test is happening no matter what reasons students offer.

The phrase emphasizes that certain factors simply don't matter to the outcome. A coach might say that playing time depends on effort at practice, irrespective of how talented someone is naturally. This means natural talent won't help if you don't work hard. A contest might be open to all students irrespective of age or grade, meaning younger and older kids all have equal opportunity to enter.

You'll often see this phrase in formal writing or serious conversations. In everyday speech, people usually say “regardless of” or “no matter what” instead. All three mean essentially the same thing: a particular factor isn't going to change what happens. When your parents say you need to finish your homework irrespective of how tired you are, they mean your tiredness, while understandable, doesn't change what needs to be done.