Baden-Württemberg the the southwestern-most state in Germany. It borders France, Switzerland, and Bavaria. It is a mountainous state and home to the Black Forest. It's the place people think of, along with Bavaria, when they hear "Germany." (Culturally, the Swabians & Franconians and Bavarians are quite different.)
Deep in the southern mountains lies the Bodensee, or Lake Constance, which touches Germany, Switzerland, and Austria. The Danube has its source in the Black Forest.
This is the state with a lot of cities you've (maybe) heard of: Stuttgart, home of Mercedes-Benz; Heidelberg; Mannheim; Freiburg im Breisgau; Baden-Baden; and some you may have not, like Tübingen. I'll start with the places I've been.
Heidelberg is a smallish university town in northeastern BW. The Neckar flows through the center, and there's a ruined Gothic-Renaissance castle up on the hill. In the castle, you can get a taste of the local wine as well as visit the German Apothecary (Pharmacy) Museum, which is actually kind of cool. The town is quaint, but there's quite a lot of tourists.
A little bit further up the Neckar, and near Stuttgart, is the similarly quaint and less touristed university town of Tübingen. There's a lovely island in the river where you can walk, and the old city walls are still partly visible, though built on. The castle houses part of the university and a museum, which is a great place to go if the weather's awful. (German weather in spring is unpredictable.) It also provides a great overlook of the river valley. The town hall has an elaborate facade. For the bibliophile, you can visit the antiquariat (used book shop) where Hermann Hesse worked.
Down in the bottom of the Black Forest is Freiburg im Breisgau. It's the quintessential Black Forest town, complete with a medieval church and timbered houses. It's the gateway to the Black Forest. I admit I was kind of bored there, but if I were looking for a getaway with gorgeous scenery, where I could relax or pop off to take a walk in the woods, that would be the place to go. It's also reportedly the warmest, sunniest city in Germany. 78 F for the average high in July is downright balmy to this North Carolinian! That's springlike for me ;)
Places I haven't been in BW: Stuttgart, where you can take a tour of the Mercedes museum and the Porsche museum and see the house where Hegel grew up, as well as your usual complement of cathedrals and castles.
Baden-Baden. If you take a guess from the name, it's a place where there are baths. Around the time Londoners went to Bath to take the waters, Germans went to Baden-Baden for the same purpose. It's still home to mineral spas and cure resorts.
Next up: Bavaria.