The Floo Network is a set of interconnected fireplaces that can take you to another location, provided it too is connected. Many wizards use the floo network to get in and out of the Ministry of Magic, as you can not apparate in and out of it. Muggle fireplaces can not be used for floo transportation, unless an arrangement is made, much like the arrangement with the Muggle Prime Minister, in Harry Potter and the Half-Blood Prince. To access the network, one must toss a handful of Floo Powder into a fireplace, walk into the heatless green fire, and declare the desired destination clearly. If it is not spoken clearly, the traveller may end up somewhere else entirely. In Harry Potter and the Chamber of Secrets, Harry doesn't say "Diagon Alley" clearly and ends up in Knockturn Alley.
Another method of using the Floo Network is to stick your head into the fire, rather than your whole body. By doing this, only your head moves to the destination and remains in that fireplace until you pull it back out. The reason this is done sometimes is that wizards will simply want to have a conversation with someone far away, but sending letters by owl is too risky or otherwise undesirable. This was done once in the books when Sirius Black wanted to talk to Harry while he was at Hogwarts, in Harry Potter and the Goblet of Fire.
Harry Potter and the Chamber of Secrets (First appearance)