An apostrophe has two jobs.
It can mark omissions, such as in contractions. Can not becomes can't. Will not becomes won't. Shall not becomes shan't. Should not becomes shouldn't. It is becomes it's or, to confuse you and make you wonder Why God Why, you could correctly use 'tis.. The apostrophe also comes in quite handy when using common abbreviations. For instance, you need to shorten the word "government" in some informal usage... it would be correct to abbreviate it to gov't.
The apostrophe can also mark the possessives of all nouns and many pronouns. So for instance, when talking about a book that belongs to Juliana, I would write: That's Juliana's book. Some people are against the possessive use of apostrophes, but it's still accepted and appropriate. If you're ever writing a paper for a truae Grammar Nazi who makes sure you know that he hates possessive apostrophe usage, then simply don't use it. You could appropriately get away with That's Julianas book in any circumstance, even if your editor/professor is not a Grammar Nazi. Just forget the apostrophe, and leave it for someone else who actually needs it. With nouns and pronouns that end in S already, the apostrophe just has to make a teeny adjustment. St. James' Church is superior usage over St. James's Church.
The apostrophe IS NOT used when indicating plurals. We all know that to make many nouns plural, we show it with an S at the end of the word. For instance: Grace has an apple, but Juliana has three apples. That is correct usage. Please, please, please... it is NOT EVER going to be Juliana has three apple's.
The apostrophe IS NOT used when adding an S to verbs. You know what I'm talking about. Verbs that you have to add an S on to show that someone is currently performing the action. Run = runs. Run DOES NOT become run's. Example: Lucy runs to the store to get some bread and eggs is correct. Lucy run's to the store to get some bread and egg's IS NEVER CORRECT so get that thought right out of your head, and smack the person who taught you that it was OK.
The apostrophe is also often used in non-English names. You'll find it in many Irish Gaelic surnames. O'Reilly. O'Brien. O'Malley. M'Gregor, shortened from McGregor (which would mean Son of Gregor). M'Kenzie (which would mean Son of Kenneth). French and Italian also use it in some surnames. De Angelo becomes D'Angelo. DeAngelis becomes D'Angelis. De Mauro become D'Mauro.
In regard to naming: This does not (doesn't) work with English names, and apostrophes most certainly DO NOT belong in first names. Ever. They do NOT indicate pronunciation. You can't turn Anneliese into Anne'Liese. Deana doesn't magically become De'Ana. Miracle is not appropriately changed to My'Rackle under any circumstances, even if Mommeigh thinks it looks cool and yooneeke. Emily massacred into Emma'Leigh makes me weep. Those are grammatically INCORRECT and make me want to tear my hair out. Every. Single. Time.
Finally, for easy reference and the love of John Lennon, right click and save this cartoon. Grammar nazi's the world over will thank you. (To see the cartoon larger, just click on it)