In general, apostrophes do three things.
They stand in for missing letters and numbers (like I love the '70s).
They help you write abbreviations and single letters in plural form (like mind your P's and Q's!).
They make a word possessive.
The apostrophe is a useful piece of punctuation to keep handy on your grammar tool belt. Right next to the comma and hyphen, of course.
Can we share some more riveting info about possession? We can? Fabulous. To make a singular noun possessive, add –'s, like this: Carmen's fruitbasket
Singular Nouns that End in S
For singular nouns that end in s already, you have two options. You can add just the apostrophe, without the s (Jess' backpack), or you can go whole hog and add the –'s (Jess's backpack).
Ultimately, it's a style issue. You should pick one way and stick with it.
You should also know that there are a couple of exceptions to the previously stated rules. (Isn't grammar great?) Words that end in –s and make an iz sound—like Moses and Bridges—should never get the extra –s. It just sounds weird.
Say it out loud: "Crazy Heart is Jeff Bridges's best film." That "idg iz iz" sound is totally weird. There are also a few biblical/mythological names like Zeus, Venus, and Jesus that should never get the extra –s, either.
Plural Nouns that End in S
For plural nouns that end in –s, simply add an apostrophe at the end to make it possessive, like this: the boys' jackets.
For plural nouns that don't end in –s, add –'s to make them possessive, like this: the men's bathroom.
For compound subjects, you have to decide if the two subjects possess something together or separately.
If the subjects share something, use one –'s, like this: Sarah and Carrie's religious beliefs. If Sarah and Carrie have different beliefs, then you would write Sarah's and Carrie's religious beliefs.
When one of the subjects is a pronoun, use the possessive pronoun, like this: Patrick's and my tree is thriving.
One last thing: as a rule of politeness, put yourself last in a list of people. It's the grammatical equivalent of opening a door for a little old lady, you polite Shmooper you.