Which is the correct 'apostrophe' to use when typing? ' (quote) or ` (backtick)

A backtick would be my last recommendation. A straight single-quote is acceptable, and a curly close quote can be substituted as an improvement. In other words, of these three:

• It`s common sense. . . .
• It's common sense. . . .
• It’s common sense. . . .

I would avoid the first, accept the second, and consider the third to be superior to the other two.

One website says the backtick:

shouldn't be used in place of the opening single quote, or for any other discernible typographic purpose

and goes on to say that the closing single quote is:

the preferred character to use as an apostrophe, as in I’m coming, or He’s with me.

It also mentions that the ASCII apostrophe:

shouldn't really ever be used in proper typography, but is often used because it's easy to type and well supported.