If you are typing, the apostrophe will usually look exactly the same as a single quotation mark (it's the same key on your keyboard). The apostrophe is used for:

1. replacing missing (omitted) letters in contracted words, e.g. doesn't, didn't, you're, etc.
2. indicating possession or ownership of objects by subjects (people, animals or inanimate objects)

We'll start with number 1 - Contracted Words:

A contracted word is one in which two words are combined to make a shorter word, usually to make speech more convenient and efficient. So, instead of saying something like, "I do not know. I will check it out for you", you can say, "I don't know. I'll check it out for you". As you can see, the apostrophe is inserted where letters are ommitted - the 'o' in do not; and the 'wi' in I will.

Contracted words are all over the place in English, and you will regularly hear them used when the words 'is' or 'are' are shortened. Here's an example (and so was that):

"Is there another project for me to work on? This one's finished now."

As you can see, the contraction of one is = one's. This could be substituted for a relevant noun, to say, "This project's finished now". The rules are the same. All you are doing is omitting the 'i' in 'is' and attaching the 's' to the noun.

You will also see this happening in words like that's (that is), but this is the singular example. In the plural, the word 'are' is contracted, so we can form words like you're (you are), we're (we are), they're (they are), and so on.

Now, to number 2 - Possession or Ownership:

If John places his bag on a table, he can point to it and say, "There's my bag" (note the apostrophe in the contraction of 'there' and 'is'). However, if someone else wanted to point to the bag and state who owned it, they would say, "There's John's bag". In this sentence, the apostrophe has a different function in each of the words in which it's used.

An exception to this rule on possession or ownership is when you are referring to something as 'it'. You might be talking about your dog, for example. Imagine there was a bowl of dog food on the floor, and someone asked, "What's that?". You could reply, "It's my dog's bowl". In this case, the apostrophe is doing the same as the example above, replacing 'i' in 'it is'; and stating that the bowl belongs to your dog.

However, the conversation could be different. Imagine you came home with your friend and your dog was nowhere in sight. Your friend might ask, "Where's your dog?". You would look around, point to the bowl of dog food, and say, "I don't know, but its bowl is still full of food!". This is the only time that an apostrophe is not used to indicate ownership. When you say the words, there is no difference; but when you write it, be aware of this exception.

A Note on Acronyms and Initialisms

There is a very common mistake made with apostrophes when it comes to writing acronyms and initialisms:

An acronym is a collection of letters, usually displayed in capitals, which substitutes for a collection of words (or maybe a sentence). Some of the most recognisable acronyms are NASA (National Aeronautics and Space Administration) and NATO (North Atlantic Treaty Organization).

An initialism is like an acronym, except we pronounce each of the letters, instead of forming a pronouncable word from them. The best known examples are: FBI (Federal Bureau of Investigation), CIA (Central Intelligence Agency), and you could even add OMG (Oh, my God!).

Both acronyms and initialisms are treated like common words, or as subjects within a sentence. If something is seen to be owned or initiated by or controlled by any of the above organisations, we would just attach 's at the end of the acronym or initialism: NASA's space program; NATO's plan to attack the rebel groups; the FBI's Most Wanted list. Note in all of these examples that it is always a lower case 's that is attached to these capitalised subjects.

A Further Note

On a slightly different note, but related to the issue of capitals and lower case 's, is a very common error we've (we have) noticed. How many times have you seen a heading like this?

Do's and Don'ts - Incorrect

At first glance, you might know what it means, but it's not grammatically correct. This heading features two plurals, of which the singular would be Do and Don't. To follow the logic of the error, in which a 's is added to the singular Do, you would need to do the same with Don't. That would look like this:

Do's and Don't's - Incorrect

We can see how awkward this is, with Don't containing too many apostrophes. The point here - a very, very important one! - is that WORDS DO NOT TAKE APOSTROPHE+S JUST TO MAKE THEM PLURAL. You should never add 's just to make a word plural. Returning to our example, this is gramatically correct:

Dos and Don'ts - Correct

Of course, this looks awkward - which is probably why people think it's incorrect - as the first word just looks like Dos (like the old computer operating system). So while the example above is correct, the following example is probably the best way to get around this problem:

DOs and DON'Ts - Correct