How do I purchase and book a round trip ticket for someone living in Ghana to come to U.S.?

As you've met them on the Internet, it should be fine to just give them the booking/ticket through the net. It's not normally necessary to have a paper ticket to fly these days (I can't think of when I last did). You just either check in online and print/have a phone bording pass, or show up to the check in desk and give them your identity documents (and sometimes a booking reference) and they take your bags, print your boarding pass and send you on your way.

So book the flight from [Ghanaian airport] to [US airport] using the airline (or online agent) of your choice, and forward them the booking/confirmation details.

However, it will not be trival for a citizen of Ghana to receive permission to visit the US. They will need to be granted a visa, which will involve completing a variety of checks. Keep in mind in this process that the main concerns of the officals deciding to award a visa or not are: will this person leave when they say they will and will they attempt to work illegally while present. It's probably not a great idea to book flights until this permission to enter has been obtained.

Edit: Be aware that some airlines apparently also require the card used for purchasing to be presented at check in (see this question). I've not personally experienced this, and one imagines that the problem is circumvented if payment is made via a travel agent.