The subjective case occurs in the *subject* of the sentence - part where someone does something.
All those pronouns are subjective.
The objective case occurs in the predicate of the sentence. One acts upon the object.
Harry snogged me.
Harry snogged him.
Harry snogged her.
Harry snogged them.
Harry snogged us.
It also occurs at the end of a propositional phrase (a phrase that begins with a preposition (with, of, to, from, above, below, by, around, etc) and ends in a noun or pronoun called the "object of the preposition.")
Harry was snogged by me.
Harry was snogged by him.
Harry was snogged by her.
Harry was snogged by them.
Harry was snogged by us.
Now, you're all saying, "MamaDeb! We know this! This is basic!"
And you're all right. I rarely see these done incorrectly. So what am I being pedantic about?
It's easy enough when you're using one pronoun. It gets tougher when you're using two pronouns. Actually, most of you get those right, too.
Harry snogged you and her.
Harry snogged them and us.
They and I snogged Harry.
The problem occurs with first person singular. It's been ingrained into us to use "I" rather than "me" in all cases. Therefore, I all too often read this in fanfics (and pro fics) and rpgs:
Draco saw you and I snogging by the Lake.
Pansy is going shopping with them and I.
It also happens with proper nouns and pronouns:
Snape caught Draco and I in the Astronomy Tower.
He danced with both Pansy and I at the Yule Ball.
And I've trained myself to, not ignore it because I can't ignore it, but to overlook it. Beta readers might well think it's proper, and rpgs are often written on the fly so there's no time to revise even if the writers know better. And, of course, they may be trying to use natural dialogue (but do we really speak that way?), but that should be along the lines "Me and Hermione are going shopping," right?
There is a trick, and it's well-nigh infallible. Get rid of the extra people in the phrase. They're irrelevent.
Draco saw I snogging by the Lake.
Pansy is going shopping with I.
Snape caught I in the Astronomy Tower.
He danced with I in the Yule Ball.
Painful, no? If you can't say the sentence without the extra people, it's wrong.
(Honesty compels me to add that this is not perfect. Properly speaking, the pronoun after the verb "to be" (unless it's helping another verb along, as in the shopping example) is subjective, so you say, "It is he." and "It is I," but that sounds stilted. It fits some characters, though.)
ETA: Okay, it's "prepositional", not "propositional", but I was asking to make a mistake of some kind, wasn't I?