Trading is the act of exchanging items between characters. In Path of Exile, this generally happens between two players. Trading is an essential part of the game, and unlike many other games there is no auction house or single monetary system (e.g. gold or coins). Instead there are various currency items (scrolls and orbs) in a 'barter economy' where players need to meet up before they can complete the transaction.
To trade with another player, right-click on his/her character and select "trade". When that player is in the same party, right-click the character's portrait on the left side of the screen and select "trade".
Although there are officially no auction houses, a large part of the playerbase uses shop indexers. These are third-party websites that search the Path of Exile forums and Public stash tab API automatically to maintain a database which players can search in order to find deals. Usually buyout prices are added to the items sold. The most popular shop indexers can be found here: List of Path of Exile related websites.
For a guide on how to set up a shop and sell items, see our Poe.trade tutorial.