Correct spelling, explanation: putting is the correct word due to a rule saying that in the case of the consonant-vowel-consonant ending of the word we double the last consonant when adding a suffix. In putting the base verb is put, so the whole word is a consonant-vowel-consonant combination, therefore we double the t. Not doubling it is a mistake, so puting is incorrect.

Definition of putting:
verb, gerund form of put, to lay sth on a surface or into sth
She was putting too many spices into the sauce.
noun, an action of laying sth somewhere or placing inside
Putting books away on the bookshelf isn’t difficult, please keep the room tidy!

Phrases with putting:
putting on airs – to act like or think to be superior to others
She’s been obnoxious with her putting on airs and looking down on everyone.
pick up what (one) is putting down – to understand what someone is saying, especially when it’s indirect and between the lines
You better stay away from this neighbourhood. Are you picking up what I’m putting down?

Incorrect spelling, explanation: this form is wrong since we should double t in put when adding a suffix (in this case -ing) and the correct spelling should be putting. This is due to the rule that when a word ends with a consonant-vowel-consonant combination then the last consonant is doubled. Therefore puting is an incorrect spelling and the correct form is putting.