To explain Phæree, I have to explain Ærth.
Ærth has three parts: Exterior Ærth, Subterranean Ærth (the Underdark, essentially), and Interior Ærth (the interior surface of the hollow world, similar to Burroughs' Pellucidar or Mystara's Hollow World).
Exterior Ærth is an alternate version of Earth. There are some extra landmasses that Earth doesn't have as well as some extra seas. Magic developed instead of technology but it's still an alternate Earth with similar animals and races. Subterranean Ærth has a much more alien ecology, since it's not based on sunlight, but it's essentially a subterranean ecosystem we could imagine existing on our own world. Interior Ærth is full of dinosaurs and prehistoric animals. It has no continents as such, but instead it has oceans that are shaped like the continents of Exterior Ærth, while the land is shaped like Exterior Ærth's oceans.
Phæree is the Counter-Ærth. It looks like Ærth, but the continents are a mirror-image of Ærth's. That means that the Phæree equivalent of Æropa (Ærth's version of Europe) looks like a backwards Azir (Ærth's version of Asia). The Phæree equivalent of Azir looks like a backwards Æropa. Every part of Phæree has an equivalent on Ærth, so there's a Surface Phæree, a Subterranean Phæree, and an Interior Phæree.
While Ærth is an alt-history Earth, but with magic and slightly different continents, Phæree is full-on fantasy. It's the home of every fantastical race of myth and legend, from elves to unicorns to djinn to oni to dragons and everything in between (although, note that the angels, demons, and gods have their own planes of existence). These creatures can sometimes be found on Ærth due to the gates that connect the two worlds, but they're native to Phæree.
Phæree is also much more chaotic than Ærth. Ærth's continents are fixed, but Phæree's are morphic and ever-changing. What is land one day might be sea the next. A vast metropolis might be a bucolic village the next day. The destinations of gates and portals might shift, and time itself might move faster, slower, or backwards. Phæree is roughly a mirror-image of Ærth, but this will never be exact.
The reason the website that Big Mac found is organized the way it is is that Phæree's inhabitants are divided into three categories: faeries, hobgoblins, and goblins. Faeries are the inhabitants of Exterior Phæree, and they include those mythical races generally considered to be benign or seelie. They might be capricious or dangerous, but generally speaking they're the good guys. Goblins are the inhabitants of Interior Phæree and they tend to be malevolent or unseelie, although like everything on Phæree they're chaotic and subject to change. Hobgoblins are the inhabitants of Subterranean Phæree (although they may wander between the interior and exterior) and they're somewhere in between goblins and faeries in nature.
There are many, many more creatures in Phæree than are mentioned on that website. Here's a list:
Faeries: alfen, alfar, aizi, asrai, azizzi, bajang, baku, barbegazi, biersals, brownies, bucca, caitshee, coatl, cushee, djinni, dobies, dragon buffalo, dragon carp, dragons (air, earth, water), dryades, dwarfs, elephant fish, effriti, elfin animals, elves, faerie animals, faeries, fantines, fays, ferrishyns, fong onhang, foo, fridian, gahes, gardsvoren, giants (earth, preternatural, wandjims), gnomes, greenfolk, grifunicorn, grigs, hippocampi, hsien, ki-lin, lamassu, leprechauns, nagumwasuk, naides, necks, nisse, nymphs, oreades, pech, pegasi, peris, phoenixes, phoukas, pixies, poppykettles, portunes, seahorses, sea lions, selkies, sheddu, shee, silkies, sirines, sphinxes, spriggans, sprites, swanfolk, sylphs, thunderbirds, tritons, unicorns, woodwoses.
Hobgoblins: achachilas, adhshees, ah at-trabs, amazons, apukus, atomies, aukis, banshees, bendithy mamau, bilwis, boggarts, bogies, boreandrills, bucentaurs, buchauns, ccoa cats, centaurs, chanques, cluracan, cobylnaus, cornoleons, dab-ihas, dactyles, derricks, djanni, dogai, dracofalcons, dragons (cold, fire, shadow), dragon elephants, dragonnes, dragon stags, dragon tigers, dragon turtles, dragon tortoises, egberes, fir darrig, giants (chenoo, firbolg, sea, stone), gigantes, griffins, grifforyxes, grims, grogach, grotesques, gwyllions, hippogriffs, hobgoblins, hu hsien, jotens, kakamoras, kelpies, killmoulises, kobolds, k'uel niu, lannanshees, leocentaurs, leoserpis, lesheys, lunatishees, maridi, merfolk, merrows, nagas (jungle, storm, water), nats, ningyo, ohdows, oliphants, onagers, opinici, pookas, poultergeists, rukhs, satyrs, scorpeoleons, scorpiontaurs, seawolves, sphinxes, shees, tengu, tiger horses, trow, tygers, uldras, undines, vermes, vetalas, weretherion, yale, yeches, yeti, zashiki warishi.
Goblins: abiku, acheri, adlets, afancs, afriti, ahuizotls, aigamuxas, airis, akathaso, als, amphisbaenas, annis, athach, bakemonos, bakrus, baykoks, barguests, basilisks, behemoths, behir, bisclavets, bodachs, boobries, boraros, brags, brollachans, bugbears, bugganes, bullbeggars, busos, caerbs, catoblepases, cerberi, ceropes, chimera, ciupipiltin, cockatrices, crottas, cyclopses, dharmapalas, deephags, deevs, dongus, drakes, drakespiders, drow, drujes, duergar, dunnies, ettercaps, ettins, formorians, fuathans, gargoyles, gaunts, ghilan, giants (acromegalians, ettins, formorians, hill jotens, mutians), glastigs, gnoles, goblins, gorgons, gorgoxen, gorska makvas, hags, hamtu-penyardin, hanon-tramp, harpies, hydrae, hydralizards, ifriti, jadian, kappas, kashas, kilyakai, krakens, kuei, lamiae, leucrottas, leviathan, losels, makaras, mandragora, manticores, margools, mimis, montegres, nagas, nagas (fire, mountain, subterranean), nhang, nightmares, night ravens, nocnitsas, ogres, ogrillons, onis, orcs, paigoels, panthuraei, perytons, pisachas, rocoserpises, sasabonsams, sea monsters, serpises, scorpythons, sluagh, sphinxes, stymphalians, supays, theriocephali, trolls, vampires, vodyaniyi, wendigoes, weretherion, will o' the wisps, willowfolk, wyrms, wyverns, yahwe zogbanus, yakhus, yeth, yowies.