I hear people say "Irregardless of the rain, I went to the market anyway" or "Irregardless of the cost, I bought the car anyway." And I always want to ask them, what does "irregardless" mean? If you look up this word in dictionary, it actually doesn't exist. The correct usage above should be "Regardless of the rain, I went to the market anyway" or "Regardless of the cost, I bought the car anyway."
"Irregardless" is not considered as a standard word since you have two negatives in one word, one is "ir" and the other is "less." I guess people use this word to emphasize how "regardless" they mean. However, it has been used so much, people think it is a correct word to use. Call me old fashioned, I have a urge to correct that person whenever I hear "irregardless."
So, for those of you who are not native speakers out there, remember that if a native speaker started using "irregardless" in his/her sentences, don't copy. It's a bad habit.