It's a lot easier to define the word epiphyte than it is to provide examples. The reason is that epiphyte is less a type of plant than it is a lifestyle, and with lifestyles, one can dabble, or one can commit. The line between regular plants and epiphytes, therefore, is kind of fuzzy.
But first things second. What epiphytes actually are, are plants that grow on other plants, generally trees. (Epi - on or above; phyte - plant) They're not parasites: they don't steal nutrients or directly harm the plant they grow on. They may get large enough to break branches off, but that's an accident, and doesn't serve the epiphyte's needs any more than it serves the host. There are a few sinister epiphytes that begin development growing on branches or trunks but eventually outgrow, overwhelm, and kill their host, notably the "strangler figs" like Ficus aurea.
A lot of the aroid epiphytes (Anthurium, Monstera, Philodendron) can either begin life on a tree branch and eventually drop roots down to the ground (primary epiphytes) or begin life on the ground and crawl until they find a tree to climb (secondary epiphytes). Either type of plant is sometimes called a hemiepiphyte. (hemi - half, partial) Some other plants can grow either terrestrially (in the ground) or as epiphytes, depending on where they find themselves, but any individual plant picks one or the other spot and sticks with it. On the other hand, some epiphytes are unable to live terrestrially, and either find themselves a branch as a seedling or they die.
Epiphytes are over-represented in the houseplant world. I suspect the main explanation for this is that epiphytes are naturally adapted to shade, so they adjust better to the lower light levels found in the home. The most common groups of epiphytic houseplants are found in the aroid, bromeliad, cactus, and orchid families.
The list that follows is far from exhaustive; I was kind of in a hurry. (This post is partly so I can be eligible for an epiphyte-centric plant giveaway Steve Asbell is doing; details on how to enter are at his post here.) I'll certainly add other plants if anybody wants to suggest some.
I haven't grown some of the plants in this list (Anthurium, Tillandsia), and one I haven't had for very long (Oncidium), so I'm not sure the recommendations and anti-recommendations are going to be all that useful, but we'll try.
I'm a big fan of Schlumbergeras. They have slightly quirky needs, but they're less complicated than I'd been led to believe, the flowers are really pretty, and the plants can last forever.
Dischidia ruscifolia doesn't do a whole lot, but as far as I can tell, they're easy to keep happy.
Aechmea fasciata should be available a lot more often than it is, because it's a very nice plant, even when it's not flowering. I find them very easy to care for, too, which helps.
(I also suspect that I would probably like Tillandsia xerographica if I got to know it, but they're rarely available here, and when they are available, they're really expensive. So I don't have one yet.)
For the anti-recommend, I'd have to go with Anthurium x 'Marie,' even though I haven't actually grown it, because I've grown a couple other foliage Anthuriums and it hasn't worked out. They're not hard to keep alive, but they need a lot more light than I can provide: without it, the leaves get either enormous and thin (A. "hookeri") or small and twisted (A. podophyllum), and neither one looks particularly good. (I haven't had a good time with my Platycerium either, but I'm not sure which of us is more to blame for that, so they might work out fine for other people and I won't discourage you from trying one.)
All epiphytic plants have certain expectations for what kind of material they're going to find around their roots, and sometimes if you have trouble with an epiphyte, the soil is the problem, not you or the plant. I've recently started adding coarse, unchopped sphagnum moss to the potting mix I usually use, in about a 1:1 ratio, for my Anthurium andraeanums and bromeliads, and it seems to have made a world of difference there. (This may be why the Platycerium hasn't worked out: for it, I used the straight potting mix, with no modification, which is possibly too heavy.) Orchids are typically grown in mixed chunky pieces of bark but are sometimes also grown in 100% coarse sphagnum. For a lot of the epiphytic cacti, my regular potting mix seems to be okay with no modification most of the time.
Most other Aechmea spp. (epiphytic or terrestrial)
some Aeschynanthus spp. (epiphytic or terrestrial)
Anthurium andraeanum (epiphytic or terrestrial)
Anthurium crystallinum (epiphytic or terrestrial)
Anthurium podophyllum (epiphytic or terrestrial)
some Begonia spp.
most Billbergia spp.
Cattleya and Cattleya alliance orchids like Potinara, Brassolaeliocattleya, Sophrolaeliocattleya, etc.
some (?) Cryptanthus cvv. (?) (epiphytic or terrestrial)
most (?) Dendrobium spp.
Dischidia spp. including D. nummularia 'Pebble Beach'
Ficus benjamina (not a lot of information about it being a strangler fig out there, but it comes up often enough that I think it must be occasionally)
Hatiora salicornioides and other Hatiora spp.
a few Hippeastrum spp.
some Hoya spp., including H. carnosa (?) (epiphytic or terrestrial)
Monstera deliciosa (hemiepiphyte)
some Neoregelia spp. (epiphytic or terrestrial)
Oncidium and Oncidium alliance orchids like Miltoniopsis, Beallara, Odontoglossum, Vuylstekeara, Wilsonara, etc.
some Paphiopedilum spp.
Philodendron bipennifolium (hemiepiphyte)
Philodendron bipinnatifidum (hemiepiphyte)
Philodendron hastatum (hemiepiphyte)
Philodendron hederaceum (hemiepiphyte)
Philodendron martianum (epiphytic or terrestrial)
Philodendron mexicanum (hemiepiphyte)
most (all?) Rhipsalis spp.
Syngonium podophyllum (hemiepiphyte)
Many but not all Tillandsia spp. ("air plants")
Vanilla spp. (hemiepiphyte)
some Vriesea spp., including V. splendens