This article focuses on the basics of Steam trading. If you're familiar with it and are looking for a quick intro to Steamexchange instead, this is the article you are looking for.
Valve introduced Steam trading back in September of 2011. You can use it to trade ingame items for games such as Team fortress 2, Portal 2, Spiral Knights etc. But more importantly, it lets you trade your games.
Unfortunately there are. You can't just trade any game you own on Steam, but only your gifts.
The ones that you haven't already opened. Once you do that they are associated with your account and can't be traded anymore.
Also, some items might not be trade-able. You can check that in your Steamcommunity inventory.
You invite someone to trade by right-clicking on their name in your friends list, and picking the Invite to trade option from the pop up menu. If the other person accepts, you'll see the Steam trade interface. You put games up for trade by selecting them from your inventory. Once you're ready to trade, you put a mark next to Check this box when ready to trade. When your trading partner does the same, the Make Trade will be enabled and you can finish the trade by clicking it.
You should always be on the lookout for people trying to scam you. Before making the actual trade, check every game you got offered to see if it is the game you want by hovering over with your mouse. Scammers will often try to trade you Gift passes or DLCs instead of full games.
You can read more about Steam trading in general on Steam support site here.