Nationalism is considered to be a kind of fundamental patriotism. This is when the people of one country (or religious, cultural, social, ideological group) have so much pride in their national identity that the protection and elevation of this identity trumps all other motivations. Communist Russia (especially under Stalin) and Nazi Germany are examples of extreme nationalism. In many cases, such as these, nationalism manifests in a reactionary government with possible implementations of racial segregation, political revolution or even international isolation. Nationalist countries will do what they have to in order to preserve what they consider their national identity.
If a country prides itself on Christian fundamentalism, that country’s nationalist individuals might endorse promoting Christianity or even legislating it as the national religion; anything to ensure the entire country represents their view of what the national identity should be. If a country prides itself on market power, they will endorse international trade, free market and cheap labor. If a country prides itself on racial purity, such as Nazi Germany, they will endorse expulsion of so-called racial inferiors. Extreme nationalists will even result to war or terrorism if they feel their national identity or way of life is threatened. Or, in some cases, nationalist groups might feel it incumbent upon themselves to promote their national identity around the world; sometimes through the use of violence. In most cases, nationalism manifests in these kinds of prejudicial and ignorant acts. It is when patriotism crosses the line to nationalism that there is cause for alarm.
Nationalism today means the feeling that people of your own nation are in some way better than people of other nations. The word "nation" here can refer to a country, to an ethnic group, to a religion, or to some mix of those. A nation is whoever a person happens to identify with -- the people who are fundamentally "like them."
Therefore, you can have Chinese nationalism today. This takes the form of Chinese feeling that their country is superior because of its growing economic and military power. Chinese nationalists are offended, for example, when other countries try to pressure them on human rights. They feel that they are better than others and that they know best. (This is not to say Chinese are the only ones like this. Americans feel the same way, for example, when pressed about the death penalty.)
As another example, one can have Muslim nationalism. You have people who identify with one another based on their religion and who feel that they are superior to others. In this case, nationalists might feel their superiority comes from their moral code.
In short, modern nationalism consists of people who identify with one another, thinking that their in group is superior to the outsiders.