Can only handle people. You can teleport yourself. But you can't teleport things that aren't actually yours. So like... your purse or your backpack can come along, but not the crate or oranges you want to sell in the next town over. Gotta take that on foot or by chocobo or cart.
(This is why the 77 caravans still make bunches of money.)
There is a hypothetical limit to the number of teleports a person can safely handle. (Back in 1.0 this was the "anima" system.) I assume that no person ever actually teleported themselves to death, though, so that's why they took away the limits in 2.0. I think it's implied that most people get pretty tired after teleporting, though, and perhaps there are some who cannot do it at all.
If you teleport together as a group only one person has to pay the teleport fee. GROUP DISCOUNT!
Teleport fees are automatically deducted from your gil balance in the game, but it's assumed you toss some coins at the guard after you arrive at your destination and they just didn't bother animating that because it'd be silly.
In 1.0 all the teleport crystals got destroyed by Dalamud causing the aether to go haywire. Most of them got eaten by voidsent, but Camp Revenent's Toll in Mor Dhona exploded into the hot radioactive mess that it still is today, and the site had to be abandoned.
Lord Lolorito personally paid for Sharlayan scholars to come by and build new aetherytes to replace the ones that got eaten by Atomos voidsent. Your teleport fees are going to him.
Because the new Scions of the Seventh Dawn refused to pledge their fealty to Lolorito, he put a statue of himself in Vesper Bay instead of an aetheryte. Giant middle finger to Minfilia.
As the warrior of light has proven, there isn't any hypothetical reason why someone can't just attune to aetherytes in other dimensions and use them to dimension hop. Or go to the moon. Or the end of the universe. (Guessing Y'shtola wasn't in her "real" body on the First so it doesn't count for her. And there's no aetherytes left on the 13th.)
Even Garleans, who can't use magic, were apparently able to use an aetheryte system, provided it had an artificial power source from cereuleum. That's wild!