Citizenship has been defined as full and equal membership of a political community. In the contemporary world, states provide a collective political identity to their members as well as certain rights.
Therefore we think of ourselves as Indians, or Japanese, or Germans, depending on the state to which we belong. Citizens expect certain rights from their state as well as help and protection wherever they may travel.
The importance of full membership of a state can be appreciated if we think of the condition of the thousands of people in the world who have the bad fortune to be forced to live as refugees or illegal migrants because no state is willing to grant them membership. Such people are not guaranteed rights by any state and generally live in precarious conditions. For them full membership of a state of their choice is a goal for which they are willing to struggle, as we see today with Palestinian refugees in the Middle East.
The precise nature of the rights granted to citizens may vary from state to state but in most democratic countries today they would include some political rights like the right to vote, civil rights like the freedom of speech or belief, and some socio-economic rights which could include the right to a minimum wage, or the right to education. Equality of rights and status is one of the basic rights of citizenship.