A good definition is here.
A good discussion of romanticism in its historical context is here.
Romanticism rebels against the Enlightenment of the 18th century, which tended to be cold and rational, gathering knowledge in Samuel Johnson's, Voltaire's and Diderot's dictionairies and encyplopedias. Enlightenment philosophers like John Locke had a mechanical way of looking at both human beings and society.
Therefore, romanticism is inherently backward-looking. (Hence the name: it draws from the classical or "Roman" era.) Romanticism is a king of paganism, preoccupied with myths, fairy tales, folklore, superstitions, magic, spirits, nature worship, etc.
Romanticism was much criticized for placing too much importance on the self and for being overly emotional. Romantics elevated beauty over intellectual knowledge, intuition over reason, childhood over adulthood, innocence over sophistication, the individual over society... These very common ideas originate from the first half of the 19th century in Europe and America.
American romanticism, typified by Emerson, Whitman, Thoreau, Melville, and Poe, grows out of transcendentalism. But there are few real philosophical differences between transcendentalism and European romanticism.
A key idea of interest to poets is Keats's concept of "negative capability." It's from one of his letters. Romantics believed in paradox and oppositions. You can find this juxtaposition of oppositions in almost any romantic poem. A typical example is Byron's "She Walks in Beauty," in which darkness and light are constantly likened. True to the idea of negative capability, this poem is full of double meanings. Even the first line can be read three ways: "she is like the night" or "her beauty is like the night" or "she walks as the night walks." Her "aspect" refers to the position of the stars in astrology but also to her demeanor. And on and on. Once liberated from rational meaning, romantic poets were able to create a tapestry of possible meanings.