The Travelling Merchant
A merchant ship calls at every city with a Port a few times a day; while it lies at anchor you sell goods to it and buy goods from it for Gold.
When it comes
The ship calls at each city on its own, so if you have more than one port, each sees its own ship. When it will come is not shown in advance. When it puts in, the bell tells you, and if you have turned notifications on, your phone or browser gets one too. The ship stays in port for a while; the Port's window shows how long until it sets sail. While the ship is in port it lies at anchor off the end of the pier on your city map; press it to open the merchant's own window. Once it has gone, it is not seen again until its next call.
Prices
The merchant takes its prices from the market: it goes by the middle price of the world's latest market trades. If there are not enough trades, or not enough players behind them, it uses the price the game sets for each good. A handful of trades cannot drag the price to extremes. The merchant sells above that price and buys below it, so dealing with other players on the market is usually more profitable. Prices are set when the ship arrives and do not change while it stays in port. Your trades with the merchant do not feed into the market price.
Trading
While the ship is in port, the merchant's window (opened by pressing the ship on the map, or Go to the merchant in the Port) shows the Gold in your treasury and the free room in your store at the top. Buy from the merchant and Sell to the merchant are two separate tabs. For each good it shows the price on that tab, how much of it your store holds and the most you can buy or sell, with where that limit comes from beneath it: your Gold, the room in your store, what the merchant has left, or what your store holds. Enter an amount or press Max; the button says how much Gold you will pay or receive. Goods you buy go straight into your store; goods you sell leave it at once and the Gold comes in at once. Goods set aside for a market order cannot be sold. The ship brings an amount of each good in proportion to the size of your store, and takes as much; the bigger your store, the more the ship carries.
When it does not come
The ship does not call at a city if you have not been in the game for a long time, if a holiday is waiting or under way, if the city is occupied, its port is held, or a battle is going on there. Nothing can be traded while you are on holiday.
The figures come from the game's own content. Balance updates can move some numbers; the rules themselves are as written here.