Citizenship is membership in a political community (originally a city but now a state), and carries with it rights to political participation; a person having such membership is a citizen. It is largely coterminous with nationality, although it is possible to have a nationality without being a citizen (i.e. be legally subject to a state and entitled to its protection without having rights of political participation in it); it is also possible to have political rights without being a national of a state - for example a citizen of Mozambique (or another Commonwealth country) resident in the UK is entitled to full political rights. See nationality for further discussion of how citizenship can be acquired, etc.
Citizenship implies working towards the betterment of the community one lives in through participation, volunteer work and efforts to improve life for all citizens.
Currently "EU Citizenship" is not quite equal in status to national citizenship (and certainly not outside the EU). Rather one holds the "nationality of a member state" and, as a result of the Maastricht Treaty, thereby becomes a "citizen of the Union". This offers certain privileges within the EU: in many areas EU citizens have similar rights to native citizens in member states. Such rights granted to foreign EU citizens include the right of abode, the right to vote in local elections and the right to work in any position (including the civil service) except for very specific positions (defense...). The EU member states use a common passport design, burgundy coloured with the name of the member state, national seal and the title "European Union" or equivalent.