What is the difference between " to smoke" and " for smoking"?

(Firstly, I stopped for smoking is not something that native speakers of English are likely to say).

I stopped to smoke

This implies "I stopped in order to smoke", where the use of to + infinitive indicates an intention:

In this case to has the same meaning as in order to or so as to. Examples She came to collect her pay cheque.

The three bears went to find firewood.

I am calling to ask you about dad.

Your sister has gone to finish her homework.

This is from edufind, where there is more about uses of the infinitive.

So you could use I stopped to smoke if you were say, walking along, stopped walking, stood still and then had a cigarette.

However, if you want to say that you do not smoke any more, you must say:

I stopped smoking

In this case the verb is followed by a gerund. In English, we stop a thing, so "stop" has to be followed by a noun, or the noun form (gerund) of another verb. Here is a link (edufind again) to a list of common verbs which are followed by a gerund (you will see that stop is listed).

"I stopped smoking" means that you were smoking, but for some reason you stopped it. For example, "I stopped smoking when my mom entered my room." Where as, "I stopped to smoke" means that you were going somewhere and stopped, so that you could smoke. For example, "I stopped to smoke on my way back home."