Germany is similar to the United States in having a federal form of government. Territorial subdivisions of the country have independent authority on matters where they have not ceded that authority to the national government.
The German word for each subdivision is Land (pronounced lont), and the plural is Laender (LEN-der is at least a close approximation of the pronunciation). Land is sometimes translated into English as "state", but some English publications use the original German, considering "state" to be an inexact translation.
Each Land has its own parliament. The head of government is the Minister-Praesident, of which the literal translation is "minister president", but the title is often rendered in English as "premier" or "governor".
Another similarity between the U.S. and Germany is that political observers watch the state elections for clues regarding the outcome of the next national election. But American elections follow a more regular schedule than those in Germany.
Most American states elect their governors on the November election day that falls half-way between presidential elections. Other states vary from that schedule, but gubernatorial election day is always the first Tuesday following the first Monday in November, of some year.
By contrast, the general elections in the German Laender are held at various times of year, and the interval between elections is four years in some Laender and five years in others.
This year, election day in three Laender happened to come up yesterday, four weeks before the general election for the federal parliament (Bundestag). Here is a New York Times report on those elections.
There are 16 Laender, and the three who voted yesterday are: Saxony (population 4.2 million, 6th largest), Thuringia (2.3 million, 12th) and Saarland (1 million, 15th). Before the 1990 reunification of Germany, Saxony and Thuringia were in East Germany, and Saarland was in West Germany.
The center-right Christian Democrats (CDU), the party of Chancellor Angela Merkel, lost ground in all three of those Laender, with the largest losses coming in Thuringia (31% of the vote, down from 43% in the previous election) and Saarland (35%, down from 47%). In Saxony, CDU support declined by only two percentage points, to 40%. (Keep in mind that, in Germany, the title of "chancellor" belongs to the head of government, not, as in Britain, to the finance minister.)
The CDU's losses were not so much to the benefit of its main rival, the Social Democrats (SPD). Two smaller parties, the Free Democrats (FDP), and the Left Party, made significant gains. The Free Democrats are classical liberals, believing in small government and low taxes. The Left Party is a combination of former Communists from the east, and former SPD members from the west, who left the SPD as it moved closer to the center of the spectrum under then-Chancellor Gerhard Schroeder, from 1998 to 2005.
Media reports caution against reading too much into yesterday's results. The CDU's largest losses came in some of the smallest Laender, with lower voter turnout than is expected in the upcoming federal election. Still, they can't be construed as good news for Merkel. She hopes to emerge from the federal election in a sufficiently strong position to govern in coalition with the FDP. That would require better results than she achieved in 2005, when she was forced to form a "grand coalition" with the SPD.
For more background on the German system, you can go to this post, this post, and this post.