English, like most languages, possesses several types of pronouns. The simplest are subject pronouns. A subject pronoun is a placeholder for a "real" noun and is the subject of the verb. In English, the subject comes before the verb and nearly always at the beginning of a sentence.
English Subject Pronouns:
|he, she, it||they|
Now, the word that takes/receives the action of a transitive verb is the object. An object can either be direct or indirect. A direct object receives the action of the verb without the need for any intervening prepositions (functioning as case markers). An indirect object has the action of the verb done on its behalf: to it or for it.
English Object Pronouns:
|him, her, it||them|
Let's look at a few example sentences.
I (subject pronoun) speak.
I (subject pronoun) hit him (direct object pronoun).
He (subject pronoun) speaks to me (indirect object pronoun).
This is fairly straightforward. The problem is that "me" has gotten a bad reputation. Somehow "I" appeals more to the common man and sounds superior to the mundane "me". Schoolteachers contribute to the stigma, perhaps not making the distinction clear to their students.
It is fine to use "me", but not as the subject of a sentence.
It is fine to use "I", but not as the object of a sentence.
The confusion seems to occur most frequently in compound subjects and objects, so let's have a look at some examples.
Harry and I chastised the monkey. ("I" is used here because it is part of the subject of the sentence).
Harry chastised the the monkey and me. ("Me" is used here because it is part of the object of the sentence).
If you hear someone incorrectly using "I" as an object, ask them if they would still use it outside of a compound object. They might (for some reason) feel comfortable saying: You can send the documents to Bill and I. But, they probably would feel awkward uttering the phrase: You can give the documents to I.
They should feel uncomfortable and, perhaps, a little ashamed at the abuse of the perfectly good object pronoun "me".