What is the procedure to invite someone from Afghanistan who can just come to visit?

Your cousin needs to apply for a visitor's visa by showing that he does not intend to immigrate to the US and that he has strong ties to his home country by showing proof of school, work, family, finances, etc. Realistically, it will probably be fairly difficult for a 18-year old from Afghanistan to show that there is something really keeping him back in Afghanistan these days.

Your cousin would apply on his own for a B-2 visa at the U.S. consulate in Kabul. If you're going to finance his trip and stay in the U.S., send him a letter of invitation (addressed to the consulate). You should give specific dates you want him to be here. Also send a completed form I-134 with your income documentation. Chances are usually small with single, young people applying from such countries like Afganistan, but you can certainly try. For more information, see nonimmigrant visa section of the website of the U.S. consulate in Kabul.