Obtaining citizenship of a country usually means that you have the same rights as somebody born with citizenship, which would usually include being able to apply for a passport; the right to reside in the country indefinitely; and the right to vote in elections. But the precise rights, and any restrictions, would depend on the actual country involved.
It depends what country you are talking about. I moved from Scotland to Sweden in 2002. Next year I will have full citizenship. I am allowed to have dual nationality and so the Swedish authorities have stamped the Swedish colours inside my British passport. I think the legislation varies from country to country.