How does nationalism influence the political and cultural development of the nation state?
Nationalism is the identification of individual(s) with a specific group, and can extend to a feeling of belonging with a nation or country. It is not always a negative thing; every person living in society feels a connection to one group/ideology or another, and usually this feeling becomes a strong identifier of who you are and what you believe.
Generally speaking, nationalism on a large scale leads to uniformity of thought, action, assumption, and bias. The most famous example are the National Socialist German Worker's Party, or Nazis, in early-1900s Germany. This group was adept at propaganda and conversion (to ideology, not religion), and was able, with the charisma of Adolf Hitler, to adversely affect most of the "normal" German peoples. In this case, the concept of nationalism -- of belonging to the German nation -- was entirely predicated on believing most, if not all, of the ideas specifically espoused by the Party itself: this is the infamous "party line" that is parroted by believers, and is an example of Ethnocentrism and Expansionist Nationalism.
On the other hand, the concept of nationalism as pride in one's country or people is the basis of most Democratic nations. In general terms, Civic Nationalism is the belief in a political system and the support of that system and its goals, with a strong emphasis on freedom of speech, individualism, and solidarity with other nations who share similar goals. Utilitarian philosopher John Stuart Mill was a proponent of Civic Nationalism over State or Government Nationalism. Most nations that practice democratic procedures fall into this category; usually, they are the result of political revolution or deliberate construction by founders.
Ultimately, nationalism is only one factor in defining the cultural development of a state. It is important to remember that nationalism will depend as much on the ideology of its proponent as on the commitment to the nation itself.
Given that not all scholars even define nationalism in the same way, there are many possible answers to your question. Please check your text and/or notes to see if there was one particular answer you were supposed to have come up with.
One way to answer this is to say that nationalism leads to political and cultural institutions that are less free and open than they might otherwise be. Nationalism can clearly do this in the realm of culture as a nation tries to protect what it sees as its cultural heritage. You then get phenomena such as the French academy whose job it is to try to ensure that the language remains free from outside influences.
Political institutions in a nation-state based on nationalism can also be restrictive and at times oppressive. We can think of how, even in a state that it relatively open like Turkey, there are oppressive aspects tied to nationalism. Specifically in Turkey, there is an effort to exclude Kurds from the political process. The more that the nation state is based on nationalism, the more this is likely to occur.