A village is a small town. The definition varies according to location, but you would expect a village to have limited facilities and a population of no more than a few thousand people. Villages are usually in the countryside.
The countryside is an area which is not built up and is generally used for agriculture. It is usually characterised by farms, fields, and woodland.
'Country' can be used as a synonym for 'countryside'. When used in this sense, it is an uncountable noun, usually preceded by 'the'. For example, "John didn't like city life, so when he retired, he went to live in the country". In this sentence, 'country' means 'countryside'.
When 'country' is used as a countable noun, it refers to an entity with national borders, for example 'France is a country in western Europe'.