Why is irrespective considered a proper word but irregardless is not?
The "ir" in "irrespective" means "not", i.e. "not respective". So "irregardless" would mean "not regardless", which would mean the opposite of what you probably hope it would mean.
Some people use "irregardless" for “regardless” but many people think it is incorrect. It is probably a combination of "irrespective" and "regardless" which is sometimes used humorously.
Oxford Dictionary says:
Irregardless means the same as regardless, but the negative prefix ir- merely duplicates the suffix -less, and is unnecessary. The word dates back to the 19th century, but is regarded as incorrect in standard English.
In addition to the prefix/suffix duplication, irregardless is used rarely. A COCA query shows 11189 hits for regardless and only 36 for irregardless. That's a reason for it to not be considered a proper word.
A proper word must not be redundant by mere inclusion of a prefix. In this case, both words make use of the prefix "ir-", which in most cases creates an antonym of the word without that prefix, e.g. something that is "relevant" is that complete opposite of something not pertinent to the matter at hand, or "irrelevant".
First examining the proper word "irrespective", we have "respective" when removing the "ir-" prefix. "Respective" carries a meaning much like "with regard to", reserving an importance that had been or will be established. Hence, "irrespective" should mean "without regard to", which it does quite nicely when the preposition "of" follows.
Now, we remove the prefix "ir-" from the widely-recognized though improper word "irregardless" and have "regardless", a word used to indicate contrast. Very often, "regardless" can stand alone without a preposition immediately afterward. This usage of "regardless" gives such synonyms as "nonetheless", "nevertheless", and "even so". When "regardless" is followed by "of", the writer states that something is true despite whatever may be the object of that "of".
The redundancy of and lack of purpose for "irregardless" becomes strikingly obvious as we include the prefix "ir-" once more, since we might note that there is no apparent difference in meaning between "regardless" and "irregardless". While English is not likely the most logical language, we should nevertheless see a sharp contrast between these two words, with the prefix causing "irregardless of" to mean "in light of" or "with regard to", which is not a documented usage of the word. However, this is the correct usage of "irrespective of", which does not duplicate the meaning of the word to which the prefix "ir-" is added.
Consequently, "irregardless" should never be used in any formal writing or even in formal speech, as it accomplishes nothing other than to weaken the meaning of "regardless". When "of" does not naturally follow, use "regardless" or one of the synonyms discussed. When of naturally follows, either "irrespective of" (my personal preference whenever suitable) or "regardless of" can substitute marvelously!