While romanticism can be applied to a time period and a range of expressions (general attitudes, literature, art, etc.), a common way to explain the characteristics of romanticism is to apply it to the literature of the time.
The previous literary time period was the Age of Reason. As the name implies, logic and reasoning were placed above all else. It makes for some cold reading. That said, some wonderful writers and works were produced during that period, including Thomas Jefferson, Ben Franklin, and Thomas Paine to name a few.
Romantic literature is a sort of revolt against the Age of Reason. Unlike the name implies, romanticism is not focused on romantic love. Romanticism instead focuses on the senses, feelings/emotions, and imagination. Another shift is in the subject matter of romantic literature. While the Age of Reason tended toward society (cities), romanticism shows a definite focus on nature, so much so that nature itself became more important or powerful than man. Art from that period illustrates this concept because while a human might be in the scene, he or she is usually not the focal point of the art nor are they painted/drawn very large.
Nature essentially became a source of divine inspiration and revelation for romantics. In a way, it's a bit New Age. They attempted to experience a "oneness with nature," to use a cliche.