Trading
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 in this 'barter economy' where players need to meet up before they can complete the transaction.
Interface
To trade with another player, right-click on their character and select "trade". If that player is in the same party, their character portrait on the left side of the screen may be right-clicked instead.
Holding Alt while hovering over a stack of currency items will display the total quantity of that currency in the trade window.
Holding Ctrl+Shift and left-clicking a stackable item in your inventory will move all copies of that item into the trade window, and vice versa.
Limitations
If an account does not have a character that is at least level 25, the player may not trade currency or unique items to other players in the trade window or put them in a Guild Stash. To circumvent this, you could drop currency or uniques in a zone in order to give them to other players.
Shop indexers
A large part of the player base uses shop indexers, most of them use the official Trading website. Indexers search the Path of Exile forums and Public stash tab API automatically to maintain a database that players can search in order to find deals. Usually, buyout prices are added to the items sold.
Guides
There is a guide on trading safely with other players.
There is a guide on making your items available for trading.