Uniswap championed the x*y = k model, also known as the constant product model. It allows a user to swap any two Ethereum based tokens. These swaps are facilitated by LPs who provide liquidity to pools. LPs currently earn 0.3% of every trade.
Uniswap borrowed the idea from Bancor, a protocol that pioneered AMM, and removed the token from design to make it more gas effective. Bancor did an ICO worth USD 150 m as opposed to Uniswap's modest USD 100,000 through an Ethereum grant.
Uniswap is the most successful DEX and leads the space in terms of volume and daily active users. It has the backing of one of the most successful research-driven VC firms, Paradigm. Uniswap, like other DEXes, is often criticised for IL. However, it has been working with researchers, and there are indications that Uniswap may provide some sort of protection from IL in the much anticipated V3 launch.