When can a village become a city?

Usually:

• The area must incorporate as a municipal corporation with defined boundaries
• The bounded area must have at least a certain number of inhabitants
• The municipal constitution (or charter, or whatever they call it) must refer to it as a "city"
• In some places the municipality must have a specific type of government, or at least one chosen from a short list

Exactly what the number of inhabitants required are, and what forms of government are allowed, varies from place to place (in the US, for example, the requirements are set by the state governments and can be different in different states).

It's certainly possible and even common for a particular populated place to legally be a "city" but referred to informally as a "town". For example, if asked I usually say that I'm from a small town even though legally it's a city, because when most people hear "city" they tend to think of places with hundreds of thousands of inhabitants, which is quite a bit bigger than the place where I grew up (in Missouri, a place with population of just 200 people can legally become a city... the place where I was born was larger than that, but still only a few thousand).