Often I hear that the confusing part for people is the "EX" of "DEX". i.e., what constitutes an "exchange".  The "D" is not without its controversy as well...as in the degree of decentralisation to qualify for being a DEX. Mostly they are all non-custodial (they don't hold your funds).

There are a number of decentralized exchanges on Ethereum that have come on the scene in the past few years. With a variety of different models, they span the spectrum of on-chain matching, orders, and settlement versus off-chain matching and order management.

Below is a table that compares competing models.


Table taken from https://blog.leverj.io/comparing-leverjs-model-versus-other-decentralized-exchanges-a0621ec84e83