The main catch for buying tickets for other people is credit card requirements: quite a few airlines require either that they present the physical card at check-in, or a specific authorization from the credit card holder (you). So if possible, it’s easier to let them pay (or book yourself using their credit card), then reimburse.
Other than that, it’s the same as booking any other ticket. You need to provide their details (passport etc), of course, and should provide their telephone/email as at least one contact point. For most airlines all you need these days is the booking reference (PNR) and matching ID, but carrying a print-out never hurts.