The Balkans is a name for a region in the southeastern part of Europe.
The Balkans are named after the Balkan mountain range located in Bulgaria and partly in Serbia. This 600 km long mountain, also referred to as the "Old Mountain", is located in the center of the region. The highest peak is Botev, 2376m, the second highest is 2198m (I couldn't find its name), and the third is Midzor, 2169m. It happens to be a site of many natural beauties, such as the Iskar river canyon.
The landmass is surrounded by the Adriatic, Ionian, Aegean, Marmara and Black seas from west, south and east, so it's also referred to as the Balkan peninsula. There is no strict definition of where the borders of the region are. Commonly, however, the following countries are considered to be on the Balkans, for one reason or another:
The region has a very bad reputation: it's said to be constantly unstable, and that it can explode into a war at any time. This is due to the violent history of the area. The Ottoman Turks invaded Europe through the Balkans in the 15th, 16th and 17th century. Austria and other European powers of the time fought many fierce battles against the Ottoman invaders on the territory between Vienna and Istanbul, where borders were redrawn nearly every decade.
In the 19th and the 20th century, there were two local wars on the Balkans, in which local powers like Greece, Serbia and Bulgaria fought mercilessly to divide whatever useful land was left among themselves -- I believe the term "balkanization" originates from this.
The almost constant turmoil escalated when Gavrilo Princip of the Serbian Black Hand assassinated Austrian Archduke Franz Ferdinand in Sarajevo on Vidovdan of 1914. This event incited the First World War.
In addition to this, the dissolution of the socialist Yugoslavia was followed by a series of wars where Serbia, led by Slobodan Milošević, attacked several of its neighbours.
In the end, even foreign superpowers had to intervene in order to stop further bloodshed in Kosovo.
This is why countries that geographically only border the crisis areas prefer not to be known as Balkan countries. You'll notice that countries such as Slovenia, Croatia, Romania, Greece or even Turkey geographically (in part or as a whole) aren't really even near the Balkan mountain -- and yet, their history undeniably binds them to the bloody Balkans.