1. Subject pronouns
Are often (but not always) found at the beginning of a sentence.
More precisely, the subject of a sentence is the person or thing that lives out the verb.
Examples:
I owe that person $3,000. – I am living out that debt. I is the subject pronoun.
He and I had a fight. – This sentence has two subjects because he and I were both involved in the fight.

2. Object pronouns
By contrast, objects and object pronouns indicate the recipient of an action or motion.
They come after verbs and prepositions (to, with, for, at, on, beside, under, around, etc.).
Examples:
The guy I borrowed money from showed me a crowbar and told me to pay him immediately.
I begged him for more time.
He said he'd given me enough time already.
I tried to dodge the crowbar, but he hit me with it anyway.
Just then, the police arrived and arrested us.

3. Subject vs Object pronouns
There is often confusion over which pronouns you should use when you are one half of a dual subject or object. For example, should you say:
*"Me and him had a fight." or "He and I had a fight?"
*"The police arrested me and him." or "The police arrested he and I?"

4. Subject vs Object pronouns
A good test to decide which one you need is to try the sentence with one pronoun at a time. Would you say, "Me had a fight?" Of course not. You'd say, "I had a fight." What about, "Him had a fight?" No, you'd say, "He had a fight." So when you put the two subjects together, you get, "He and I had a fight." The same rule applies to the other example.
You wouldn't say, "The police arrested he," or, "The police arrested I."
You would use "him" and "me."
So the correct sentence is, "The police arrested him and me.

5. To sum up…
Subject pronouns are those pronouns that perform the action in a sentence. They are I, you, he, she, we, they, and who.
I make cookies every Sunday for my co-workers.
In this sentence, “I” is the actor (subject pronoun) performing the action of making (verb).
Object pronouns are those pronouns that receive the action in a sentence. They are me, you, him, her, us, them, and whom. 
I give them cookies every week.
In this sentence, “I” is the actor (subject pronoun) performing the action of making (verb). “Them” is the noun receiving the giving; it is the object.