The word "it's" is a contraction of "it" and "is". Remember what I said about contractions? Smush the word together until one or more of the letters collapses under its own weight and becomes the apostrophe. In this case, "it is" becomes "itis", then "it's".
The word "its" is a possessive pronoun. The object belongs to "it". In English, the pronoun "it" is used for objects without gender. This problem does not arise in Spanish, where everything has gender, but this blog is about the English language. If an object owns another object, then you use the possessive pronoun "its" to show that possession. For example, the tires belong to the car, so you could write, "The car spun the car's tires". However, the word car is used redundantly, so you’d want to write, "The car spun its tires." The word "its" is complete without an apostrophe. Never put one at the end to show possession; the word itself shows the possession.
How to remember which to use: Say the sentence out loud, replacing the "it's" or "its" with "it is". If it makes sense when you make the replacement, then you want the contraction with the apostrophe. If the replacement doesn't make sense, then you want the possessive pronoun without the apostrophe.
Going back to the car and tires, with the replacement phrase, you would say, "The car spun it is tires", which makes no sense, so you know you want to leave the apostrophe out and make it possessive. The tires belong to the car. However, you can safely use the contraction in the sentence "It's making a lot of smoke," because the replacement phrase makes sense there. "It is making a lot of smoke."
Either way you look at it, when the car spins its tires, it's making a lot of smoke.