Naturalization is when a person changes his or her nationality. Most people have the nationality of the country where they are born and grow up. In other cases, one can be a citizen of a country because the individual in question had a parent, grandparent, etc who was/is a citizen of a different country. If they move to another country they will still keep their own nationality at first, but if they stay there for many years and want to stay there for the rest of their lives they may want to change their nationality.
Different countries have different laws about naturalization. Usually the person must make a promise in a court of law. He will have to promise to be a good citizen and obey the country's laws. Sometimes a country gives citizenship to someone who hasn't asked for it. Sometimes, someone can keep their first nationality as well as getting a new one. This is "dual nationality". Sometimes children whose parents have different nationalities are allowed to have dual nationality.
Someone who has been naturalized will normally have all the rights that any other person belonging to that country has.
The reverse also happens. States sometimes remove citizenship. This denaturalization can leave the former citizen stateless.