Basically, you join payments together to 'anonymize' coins, while trading coins for other coins to make proving someones ownership of a specific coin harder.

Using Joinmarket as an example;
There are makers and takers in the Joinmarket network. Makers run a script that offers their coin for coinjoin transactions in exchange for a small fee. They sit around on the network, looking for a taker who needs coins mixed.

Takers, on the other hand, pay the makers to trade coins with them. Takers will come into Joinmarket announcing that they want to mix a certain amount of coins. Eventually a maker will accept his offer, and the maker and taker (and possibly more parties) all make one big Bitcoin transaction with lots of outputs to different addresses.

Joinmarket is just one of many ways a coinjoin can happen, but this should give you the general idea. 

The whole taker-maker process aims to mix the coins for the takers and the makers, making it harder to tell who owns which coins after the coinjoin-transaction. It will be hard to tell who now owns each output, if someone owns several outputs etc., which is the end goal of the transaction.

Users can run this process as many times as they like, the assumption being that more coinjoin transactions = better privacy.