Think of it like this
A factory, which creates and stores address of contracts which corresponds to each token pair
A router, while calling swap refers to this factory and selects correct pair contract
A pair contract, which basically have 3 functions
Mint - to mint lp tokens while liquidity is added
Swap - To make the actual swap and give token A/B to the caller
Burn - To accept lp token and send back corresponding token amounts

In general, a router routes the transaction to pick correct pair from a given factory and perform operations