Dai is an ERC20 token on top of Ethereum. It might be more accurate to say Dai is a stablecoin coupled to MKR (MakerDAO), a governance token. Dai is over-collaterlized.
https://makerdao.com/en/ https://developer.makerdao.com/dai/1/
Terra is a coin with ticker LUNA, and is coupled to a stablecoin UST. As I understand, LUNA/UST was under-collateralized and could not handle what was essentially a virtual bank run.
Tether is a centralized stablecoin that is also reported to be under-collateralized, but it has bridges and industry connections to the wider crypto market which has kept it from crashing and burning so far.
More importantly, DAI is overcollateralized using ETH and other coins. UST was algorithmically pegged to the USD with an implicit backing by Luna. DAI still technically has a depeg risk (e.g. if ETH has a flash crash of > 50% that it doesn't recover from) but the risks are much lower. It's _probably_ safe in the long term although my stablecoin of choice is USDC.