If a new cartridge does not work, take the cartridge out and examine it. (If an inkjet, shake it near your ear and make sure that there is liquid ink inside.) Make sure the 'print head' (where the ink comes out) is not clogged or otherwise unable to let the ink out. Sometimes the protective strip leaves residue and needs to be cleaned. Use clear 100% alcohol with a cotton swab to gently wipe the area. When the cotton swab comes away with ink on it, you'll know it's clean. Sometimes the ink itself dries and clogs the print heads. When this happens, you can set the cartridge in a quarter inch of warm water to allow the ink to soak off and the print heads to become unclogged.
Also examine where the ink cartridge meets the electronics. You can clean this gently with a pencil eraser. Do the same thing on the inside of the printer - again, gently. All you want to do here is push away any corrosion. Be sure to wipe away any eraser residue, too.
Lastly, when a new (or used) cartridge is replaced, always run the printer cartridge alignment software which comes with the printer.