"There are a lot of people in this country."
Translation:В этой стране много людей.
As BenYoung84 says, not only sometimes.
In addition to indicating possession (which is the origin of the name genitive), Russian takes genitive after a lot of "quantity indicators"
Following cardinal numbers (genitive sg. when following numbers ending with 2, 3 or 4 but gen. pl. after 5 and higher...except e.g. 23 and 854 which are ending with 2/3/4...)
After e.g. много/немного, мало, стакан/бокал (i.e. a glass of something, as a quantifier), бутылка, грамм, сантиметр, ...
If you intend to say "some" (aka an indefinite quantity of something, like French 'de'), EVEN if you omit using a word like "some" for the quantity. Examples: (a) Купил немного хлеба! (b) Купил хлеба! (In this example "some" is omitted, but there's still an undefined quantity there, hence хлеба in gen. sg.)
Finally, genitive follows нет and negated transitive verbs (meaning verbs that take an object). Example: Я не пью воды (genitive, as opposed to Я пью воду which takes the object in accusative).
Actually, Я не пью воды is not how we say it nowadays. It would have been used that way in the 19th century.
Today, transitive verbs will just use the Accusative, with some important exceptions for more abstract verbs/objects, verbs of existence and sometimes verbs of perception (e.g., не иметь возможности, не обращать внимания, не видеть смысла).
Иметь, in particular, consistently switches to the Genitive, just like in the days of old (except in иметь в виду что-то, which uses Accusative even when negated).
этой is feminine singular prepositional, этом is masculine singular prepositional.