Subject vs. Object
In grammar, a subject acts, and an object is acted upon; in English, we have different pronouns to signify the distinction between those grammatical categories. In other words, "I" is a subject while "me" is an object while "my" shows possession; "we" is a subject while "us" is an object and "our" shows possession. This shifting of pronouns (and nouns) depending on the slot the word fills in a sentence is called declension. (Not all languages bother with all these distinctions. Chinese, for example, does not decline its pronouns in this way: the singular personal pronoun, transcribed as "wo," is always "wo," whether it's a subject or an object; to make the personal pronoun plural, you add a uniform ending to "wo" and have "a bunch of I's," or "we." It makes Chinese grammar very easy to master. What makes Chinese hard to learn is its intricate, elaborate writing system and the fact that it is a tonal language--tones can be very hard for a Westerner to distinguish, memorize and pronounce correctly.)
Very small children often have a problem with the concept of declining pronouns--hence a toddler might see a cookie and declare, "Me want that!" Older people, however, often manage to master the idea--up to a point. People want to sound smart, and they've been told that it sounds smarter to say "he and I" than "him and me"–and certainly that is true if we're talking about subjects: if you informed your listener that "Him and me went to the movie," you'd sound like an illiterate yokel who couldn't find the exit sign after the film. So some people have fallen into the habit of ALWAYS saying "he and I," even though there are times when the phrase "him and me"is exactly what is needed.
Example: "Mom bought an ice cream cone for he and I."
You wouldn't say, "Mom bought an ice cream cone for I."
You wouldn't say, "Mom bought an ice cream cone for he."
You would say, "Mom bought an ice cream cone for me."
You would say, "Mom bought an ice cream cone for him."
Thus you would say, "Mom bought an ice cream cone for him and me."
The worst violation of this rule was committed by a friend of mine who, when discussing problems in her relationship with her boyfriend, would say, "I'm really worried about Mike and I's relationship."
That's right: Mike and I's relationship. Hearing that made me feel like my brain was a pat of hard, crumbling butter being scraped over burnt toast with a bent knife. In other words, it grated. I'd correct her as gently as a I could--she was, after all, an English major, an aspiring poet who claimed to love nothing in the world (not even Mike) as much as she loved words. Her grotesque abuse of certain words was not the reason we stopped being friends, but I admit that after our falling out, I took comfort in the fact that I'd never have to hear about "Mike and she's relationship" ever again.