Between 1795 (Third Partition of Poland) and approximately 1916 (World War I) most of the territory of modern-day Lithuania was occupied by the Russian Empire, with south-western part of the country remaining under control of German-speaking Kingdom of Prussia.
During that time Russia has imposed its own administrative divisions over the region, which are reflected in specification of birth, death and marriage places, making their understanding important for genealogical research.
For additional help with matching the old birth place name to a modern location, see Finding places in Lithuania.
The largest unit of administrative division of Russian Empire was "губерния" (gubernya), most commonly translated to English as "governorate". Territory of modern Lithuania was divided between 3 governorates, named after their corresponding principal cities: Kovno/Kowno (Kaunas) governorate, Vilna/Vilno/Wilno (Vilnius) governorate and Suvalki/Suwalki governorate. The fact that (in English) the name of the governorate and the name of the principal city are the same is the source of constant confusion, as the word "governorate" (or other administrative divisions mentioned below) was usually omitted from foreign records. Thus, if someone claims to be born in "Kovno", it is impossible, without additional information, to tell whether they were actually born in the city of Kovno (Kaunas), or the corresponding governorate, which covered significant territory and included hundreds of other towns and villages.
New! A transcribed list of inhabited places of Suwalki governorate is now available.
The following map shows the boundaries of 3 governorates relative to the boundaries of modern Lithuania, in the second half of 19th and beginning of 20th century:
A peculiar thing to note is the location of Kaunas within the Kaunas governorate. As it's right on the border, there are places just a few miles south of Kaunas, which, nevertheless, belonged to Suwalki governorate.
Governorates were further divided into smaller units called "уезд" (uyezd). As Uyezd Wikipedia article states, this is something similar to a "county" in English-speaking countries (you can then think of governorate as something being equivalent to a "state" or "province"). Again, uyezds were named after their principal cities. Below is a complete list of uyezds for relevant governorates at the end of the 19th century, along with a Russian name and modern Lithuanian name of its principal city (also shown on the map above, together with corresponding uyezd boundaries):