using "-in'" instead of "-ing" is a stylized and casual way of writing which mimics many dialects of English which drop the "g" while speaking (or speakin', in this case) Whenever I'm casually typing to my friends, I always drop the g because it's closer to how I casually speak Example: Proper -------"Hello, John. I was thinking about walking to the store briefly in order to pick up some milk. Did you need me to pick anything else up while I am already out?" Casual -----"Hey man I was thinkin about walkin down the store real quick to grab some milk. need anythin else?" It is important to note that this is not used within the boundaries of proper writing conventions. If you are unsure if you should be using -in' or -ing, simply use -ing. Hope this helps :)

Stick to correct English while you are learning. By the way, the tendency to drop the "g" is found mostly in the speech of Americans. It is not common in British English.

The "in' " form is only used in certain dialects. It's only written in quotes, subtitles or song lyrics, but otherwise it's never written. A lot of native speakers just consider it a mis-pronunciation made by uneducated people, but some people might think it's just more informal. Either way, it's definitely non-standard.

I can be wrong as English isn't my first language, but I suppose, -in' is a slang thing. Like wrong pronunciation, because you cannot pronounce "-ng" sound correctly.