What does it mean to use a single quote to abbreviate things?

Solution 1:

Thanks to the OP for amending his question to

how do I know when to use them? Or, when I saw them, how do I know what they stand for?

Unfortunately, as with so much in English (or any language really), a lot of it is just observation and rote practice.

It might help to remember that your first two examples are (somewhat) representative of elision in slurred colloquial speech. Jammin', or anything ending in -in', is pretty clearly an elision of -ing. You could use this for any verb and be well understood.

O' is a little less clear. It seems to have stronger historical roots (e.g., the luck o' the Irish) but in contemporary usage it represents of slurred into /ə/ (schwa). If I actually pronounce it /ow/, it's usually for comic effect such as the Irish example or in various cases of hipster irony.

'Cause is just a fairly standard colloquial form; I can't think of any pattern that applies to anything similar.

Can I create omissions arbitrarily?

As long as they're intelligible, sure.

But your next question is going to be how to tell if they're intelligible. Then we're back to where we started: Make a note of helpful patterns, listen for how native speakers elide certain sounds in speech, and experiment just to see if anyone understands you. :)

Solution 2:

In each case the apostrophe stands for omitted letters. Is your confusion because your three examples involve omission of letters at the beginning or end of a word whereas can't and you'll omit letters in the middle of the contraction?

The priceless Purdue OWL site draws no distinction:

• shouldn't = should not
• didn't = did not
• could've= could have (NOT "could of"!)
• '60 = 1960