FOR ALL INTENTS AND PURPOSES is the usual and proper form of the phrase meaning “in every practical sense.” It dates back to English law in the 1500s, originally cited as “to all intents, constructions, and purposes” in the Oxford English Dictionary as early as 1546, and written “to all intents, constructions, and purposes” in an act adopted under Henry VIII in 1547. The “for” instead of “to” came later, at least in America.
The phrase is sometimes misconstructed as “FOR ALL INTENSIVE PURPOSES.” The cause of the confusion is rooted in this phonetic similarity; that is, if you were to say these two forms out loud it might be hard to tell the difference between the two.
Thus, the incorrect “for all intensive purposes” is what is known as an eggcorn, a label invented in the early 2000s by linguist Geoffrey Pullum to describe words or phrases that are misheard and consequently reform into a new word or phrase. The difference is clear when written down, but when spoken, the two words sound very much alike.
You should opt for the proper idiom, “for all intents and purposes,” over the nonstandard alternative. This is especially important in your writing where the alternative is not considered correct.