Because Romantic Nationalism actually combines two historical concepts, it is important to look at what simple "nationalism" is first.
Nationalism is the concept of identity mainly through country. It can be contrasted with the concept of patriotism that depends on the country's leadership and the decisions that are "good" for the people within that country. Nationalism makes no such determination. In nationalism, identity with the particular country is enough.
Now to go on to the meat of your historical question in regards to Romantic Nationalism. Specifically, the term "romantic" can be quite confusing (making advocates of literature think of the concentration on feeling and nature). So, please realize that Romantic Nationalism also goes by a couple other names that aren't quite as confusing: identity nationalism and organic nationalism. All three of these terms simply refer to the tendency to identify with ones country because of the unity it inspires in its people. Further, this type of "organic" identity puts a specific focus upon people who are truly born in that country, and not "assumed" into that country by immigration or other means.
In regards to "when" and "what" of Romantic Nationalism, "the period" of Romantic Nationalism would be the eighteenth and nineteenth centuries. You can see Romantic Nationalism hidden deep within the French Revolution, for sure. The cultural aspect and pride in one's nation can be seen in the focus upon the arts, language (even colloquialism), and folk culture of the country.
Take a look at Fichte's "Address to the German Nation" as a good example:
The first, original, and truly natural boundaries of states are beyond doubt their internal boundaries. Those who speak the same language are joined to each other by a multitude of invisible bonds by nature herself, long before any human art begins; they understand each other and have the power of continuing to make themselves understood more and more clearly; they belong together and are by nature one and an inseparable whole.
Note the concentration on the unity of the nation here. The people in the country are united by "language" and "invisible bonds" from "nature." In my opinion, this is where the "Romanticism" part of the term National Romanticism comes in. Above, Fichte applies these two ideas (language and natural bond) even before the cultural bonds of art and literature. It is language and innate bonds that make the people of one country "an inseparable whole."