Look, let's take this step by step, shall we?
photo by Tim Green
To regard means to notice and consider. So, therefore, regardless means not noticing something and considering it.
He went out in his thin jacket regardless of the blizzard.
So, how about this monstrosity?
He went out in his thin jacket irregardless of the blizzard.
Well, usually ir- means means not, as in irrational or irreversible, in which case the line above makes no sense at all. Just sometimes, though, ir- means to cause to happen, as in irradiate or irrigate. As irregardless seems to be an attempt to make the word regard stronger by a double negative this last meaning is, admittedly, a bit nearer to what I think is meant by irregardless - but it's still not the same.
But does irregardless work as an extra strong-version of regardless?
Well, if it does, only at the expense of making the speaker or writer look woolly-minded and illiterate. So, personally, I avoid it.
Word To Use Today: regardless. The French word regarder is one of the first I ever learned. It comes from re- which means again, plus garder, to guard.