It varies, there are a bunch of different types of stabletokens. The two main approaches I'm aware of are:
Tokens that are issued and backed by a trusted third party. Tether, for example, issues one USDT token for every USD that is deposited with Tether Inc. and you can redeem USDTs for USD again any time. I'm not particularly fond of this approach, but it's simple and popular and as long as you're not holding USDT long-term I don't see a big problem with it as a day-to-day currency. Just make sure the issuing company is audited and you're prepared for the possibility that they could turn out to be lying.
Tokens that are issued by on-chain smart contracts, backed by other digital assets. DAI and Liquity are examples of these. They are more complicated but IMO the better choice because you don't have to trust anyone - you can see the token's backing right on the blockchain itself and know whether it's actually worth what the stabletoken needs for support.
One of the nice things about the on-chain smart contract stabletokens is that they can be backed by less-stable tokens, such as Ether itself, so you can get the best of both worlds out of them.