The standard conception of skeletons and zombies in roleplaying games these days is strongly informed by the Dungeons and Dragons approach. In this model, skeletons and zombies are the weakest undead, raised as mostly mindless servants by (primarily) evil necromancers.

This is hardly the only way to use them, of course. In the Fifth edition of D&D, it is strongly implied that these undead are animated by specifically evil forces, and are evil themselves. If left uncontrolled, they will strive to kill any living thing they encounter.

The term skeleton obviously just refers to the collection of bones from a formerly living creature, in this case now undead, or at least re-animated (that's right, those don't have to be synonymous). There's a ton of these throughout folklore and myth, but almost none are servitors to evil wizards. In Japan, there are not only undead skeletons, but the sinister Gashadokuro, a giant undead skeleton made of the skulls and bones of men slain in battle. They eat people, and seem entirely free willed, if not truly possessed of thought. They can be invisible, and can only be defeated by holy charms. They return on their own, like any haunting. Likewise, the skeletal woman known as the Hone-onna, who returns from the dead to have sexual relations with someone she loved in life, much as Eastern European vampires were said to do. In some myths from the natives of the American Southwest, a skeleton might return from the land of the dead just for some tobacco and a chat, and like European folklore, Death itself is often portrayed as a skeletal figure, though not always fearsome. In some German tales, the dead gather just to dance, usually in skeletal form, or even use their own skeletons and skulls, or those of others, to play 'ninepins' or bowling. In general, a skeleton returns as any ghost might, as a haunting, simply often one with corporeal form, a dead person who doesn't want to stay dead. The necromancer of modern pop culture doesn't enter into it.

As for zombies, well, modern pop culture has melded a lot of things together. Like the skeleton, the dead rising for their own reasons is very common across the world, sometimes hungry for flesh or blood, like the german Nachzehrer or the Arabian ghul. But these are more reasonably termed ghouls or vampires depending on their diet. Zombies as we know them, a necromancer's corpse servants, comes almost entirely from the Voudon traditions, where they were slow, enslaved creatures used as slaves for menial labor. The horror was becoming one, not fighting them. This got combined with the ghoul's appetite for flesh to make the modern notion of the zombie.

All that said, what cool things can we do with these most basic of undead?

Obviously, the easiest hack is to make them spontaneous, free willed. They either hunger, or they come back like revenants for their own reasons, and must be dealt with as one would a ghost, even though they have bodies. Violence might not even be an option. They'd also tend to be solitary, unless you happened to run into a group that died together. As they came back of their own will, they might be mindless, but more likely, they will know exactly who they were, and retain memories, skills, classes. (Obviously that's what a skeletal champion is, but they could be any class or work in any system, and might not even be evil.)

Another interesting approach is having undead bodies specifically animated by fiends, which tracks pretty well with a lot of European folklore where 'the Devil' brings someone back from the dead via possession. Sometimes they retain the original memories, but sometimes not, and the soul has either been replaced or subjugated by the fiend inside. In some worlds, possession of the living or the dead might be the only way a fiend can manifest in the physical world, and that changes how both the undead and the fiends themselves must be dealt with, possibly necessitating someone keeping it busy for a long exorcism..

This works with 'undead' actually animated by aberrations or Lovecraftian monstrosities as well.

You can also work very easily with re-animated corpses that aren't really undead. The animate objects spell from D&D can raise a corpse for work or fighting without all that evil necromantic power being involved. A druid or plant mage in any setting or system could infuse corpses with fungi (like the Spore druid) or vines, and then just control the plant or fungus. Not necromantic at all, really, but still scary and a bit gross. High tech could bring back the dead as workers or warriors. And so on. (I really recommend GURPS Undead and GURPS Zombies as resources for any game involving this sort of thing.) If you have a variety of origins for the walking dead in your game, even basic skeletons and zombies can be scary, novel, and interesting, and take research to deal with permanently.