Alternative spellings of the name of this state are Baden Wuerttemberg and Baden Wurttemberg.
Baden Württemberg consists of two parts, Baden and Württemberg. The former is subdivided into the Regierungsbezirk Freiburg and the Regierungsbezirk Karlsruhe, while the latter consists of the Regierungsbezirks Tübingen and Stuttgart.
Here are the different regions for each Regierungsbezirk:
Among the West-German states, Baden Württemberg is one of the youngest, having been founded in 1952 through a unification of administrative areas that, until the end of WW I in 1918, had been mostly covered by the kingdom of Württemberg, the grand-duchy of Baden and the kingdom of Hohenzollern. (Of course, the actual twists and turns of local history are far more complicated. See the wikipedia entry  on the state). The consequence of this - and that's the important bit a traveller should know - is that there are now two tribes living together in the state: Badener in the west and Schwaben in the east. Both speak different dialects (see below) and share a love-hate relationship towards each other that's nurtured with a lot of humour. For what unites both tribes and the rest of the people living here is a pride for "their" Baden-Württemberg and what they have made of it since its creation, that's surprising for Germans from up north. Since 1999, the state has been advertising itself all over Germany with the slogan "We can do everything, except for speaking German." (Wir können alles, außer Hochdeutsch), a tongue-in-cheek play on the infamous dialects (see below).
And indeed, Baden Württemberg is doing quite well in terms of economics compared to other places in Germany. It boasts the lowest unemployment rate of the Federation, some of the best universities in Germany, a GDP per capita that rivals Switzerland and is the only German state that still has a higher birth than death rate. The European Statistics Office (Eurostat) has called Baden-Württemberg the "high-tech central of Europe". And, famously, the percentage of people owning their own home is by far the highest in Germany.
The main reason for all those superlatives lies deeply in the history of the land: Although nowadays there are about as many protestants as catholics living in Baden Württemberg (and a third group of comparable size without religous faith), during the reformation South-West Germany was strongly influenced by the schools of Martin Luther, John Calvin and Huldrych Zwingli, which left behind a society with moral values circling around hard work, self-control and the general motto "God helps those who help themselves".
Hence the country that was once dirt poor, having to struggle with hard winters and frequent famines, today is plastered with high technology companies. The most important sectors are mechanical engineering (most famously Robert Bosch Inc.), Chemistry (BASF in Ludwigshafen), Biotechnology and, above all, Automobiles (which were, in fact, invented here, as everyone will be happy to point out). Daimler Chrysler and Porsche were founded and still have their headquarters around Stuttgart; Audi, Volkswagen and others have large plants in the state. If one counts in the small and medium-sized suppliers, every other employee in Baden Württemberg is working for the car industry, directly or indirectly. As Max Weber, a philosopher at Heidelberg University said, around here, it's "Capitalism as it was meant to be".
People from Northern Germany (who generally consider themselves more relaxed and outgoing) will fret at such bland numbers, but most people in Baden Württemberg are quite happy with things as they are: Since the foundation of the state, the same (conservative) party has always been in power. If you were searching for the stereotypical Germany the world jokes about - disciplined, hard-working, introverted, correct and stiff - you will find it well alive in Baden Württemberg.
While every region in Germany has its own dialect to the formal language (Hochdeutsch) Baden Württemberg (together with Bavaria and maybe Saxony) is among those regions where the difference between formal German and the local dialect are strongest, even to the point of being incomprehensible for native German speakers from further north.
The two most prominent dialects spoken in the state are Badisch and Schwäbisch, but there are numerous others (among them Allemannisch, spoken by the tribe that earned the Germans their name in French, living around Lake Konstanz, Kurpfälzisch, spoken in the region surrounding Mannheim and Heidelberg and Fränkisch in the north east). Needless to say, the differences between those languages, albeit fundamental issues of local pride for the natives, are very difficult to hear for the outsider.
In stark contrast to other areas of Germany (where dialect is considered to be the language of peasants), people in Baden Württemberg, especially outside of Stuttgart, tend to be surprisingly proud of their dialect and might even be reluctant to speak formal German with foreigners (which is no problem for the English speaking visitor, of course. As everywhere in Germany, even in remote villages you will always find someone around who is able to speak at least broken English). A Linguist might point out that Schwäbisch and Badisch are in fact enhanced versions of German (the term Hochdeutsch actually means "Southern German". It was spread over Germany in a period of the south's hegemony and the actual south German dialects moved on from there). But a more realistic explanation for the reluctancy of many southerners to speak Hochdeutsch might be that many of them just don't know how to do so properly.
Stuttgart has an international airport which is served by all major carriers. Frankfurt international (FRA), the busiest airport in mainland Europe is, although not in Baden Württemberg, well within reach by train (1 hour from FRA to Stuttgart main station via high speed ICE connection). Low-fares airlines offer services to the local airports of Karlsruhe-Baden Baden and Friedrichshafen.
Travellers beware: "Frankfurt Hahn", the big hub for low-fares airlines should not to be confused with FRA. In stark contrast, it has no train station and lies rather remote. It is still possible to get from Hahn into Baden Württemberg rather conveniently, but it sure takes a lot longer and more hazzles than from FRA.
All major cities are well connected through the Deutsche Bahn rail system. Ulm, Karlsruhe, Mannheim, Heidelberg, Stuttgart and Freiburg even have ICE connections (slick, comfortable, white high speed trains travelling at up to 190 mph (300km/h)). Tickets can be booked via the Deutsche Bahn Website 
Baden Württemberg (as well as some other regions in Germany) offer a special train ticket (Baden Württemberg ticket). It is valid for one day. With this ticket up to five people can use all regional trains within Baden Württemberg for 30 Euro alltogether. That means you can use all trains except InterCity(IC), InterCityExpress(ICE), EuroCity(EC) and some special trains. As it is valid for up to five people you can ask others, if they want to get to the same direction and share their ticket.
Baden Württemberg has an excellent rail network, serving even quite remote areas. Especially rural villages are served by busses which generally leave from main train stations in larger towns and cities.
Of course you can always use your car. If you are travelling in the Black Forest or the Swabian Alb during winter, bring snow chains as some smaller roads may not be plowed frequently enough. When travelling on the Autobahn, the same precautions as everywhere on German high speed roads apply: If you're not willing (and prepared) to drive consistently above 100 mph (160 km/h), stay on the right. Make room for people trying to overtake, use your common sense, don't drive faster than you can think.
For those interested in "high culture":
For those fond of nature:
For those interested in touring castles
The official Ba-Wü tourism homepage is at http://www.tourismus-bw.de/. Click on the "English" link at the top.
Baden Württemberg contains some of Germany's most significant wine-growing regions. Much of the wine economy is in the hands of local co-operatives and the locals enjoy the wine in old-fashioned wine cellars. The best wine grows in an area called the Kaiserstuhl in Baden.
Fruit brandies, e.g. Obstler (distilled from apples and pears) and Zwetschgenwasser (plums) are just two of the most common spirits. The queen of Schnaps is without any doubt the Kirschwasser (also sometimes referred to as Kirschwaesserle) made out of cherries from the black forest area. These are commonly drunk after a meal in a restaurant.
There are some breweries of note in the region, of which Rothaus is one which enjoys cult status and brews one of the best beers in Germany.
Baden Württemberg is one of the safest regions in Germany. In large cities like Mannheim and especially Stuttgart, be aware of theft. Other regions are safe and you can travel alone without any problems. Even walking alone late at night is no problem.