Most writers on lieder locate the genre as a late development of folk song or Minnesinger troubadour lyric, but a much better working definition (for the beginner at least), is "minimalist chamber opera".
That is to say, the music isn't minimalist, but that the number of singers is minimal (one or sometimes in a great while, two), the accompaniment is minimal (one piano), the singing is operatic in nature (but usually scored more towards mid-range than extremes of top or bottom) and there are no costumes or sets involved. (Although I suppose you could...but why?) The songs are short, three or four minutes tops, and singly or arranged in a song cycle usually are lyrical in form but narrative in content: the very best of this genre welds superb poetry, a broodingly introspective storyline and graceful, expressive melody into a seamless whole greater than the sum of its parts.
Lieder, sonatas, and similar pieces for small groups florished to a great degree in the Biedermeier era of the Hapsburg empire: (1818-1848). In the wake of the French Revolution, Prime Minister Metternich instituted strict censorship over the arts, a secret police to ferret out seditionaries, and a large-scale beaurocracy to handle all this, which, a century on, inspired Franz Kafka. Accordingly, most social life and artistic endeavor turned inward-looking and focussed on psychological studies, dreams and relationships. Instead of problematic religious and/or historical themes, artists painted single or group portraits (especially the genre known as "conversation pieces"), still lifes, and landscapes. While British Romanticism included biting social satires by Lord Byron, Percy Bysshe Shelley published a defence of atheism, and William Blake hinted that the angels were not the only wise, German Romantic literature tended towards ghost stories and poetry about love, lost love and nature, usually involving well-bred well-read young men with clinical depression, death, madness, severe weather and/or Alpine scenery contrasted with lissom, pure, young maidens of tragically inappropriate station, sunshine, spring, flowers, wholesome family living and the innocence of children.
Home life, not public display, was the center of all life: it became fashionable to "live with your children", eating, playing games, and homeschooling together instead of hiding them in the nursery, more people stayed home rather than went out, and houses, furniture, and fashions were often described as being "homey" or "cozy" -- apartment buildings of the time were planned so that the owner could have his shop or office on the first floor, live (comfortably) on the second, and get further income by renting rooms or warehouse space in the cellar and upper floors, so that no one need leave the house. (Clothing was at the time, usually supplied by an at-home seamstress or visiting tailor, other goods were sent "on approval", and it was assumed that the cook knew her duties.) At the same time, aristocratic manners (either Hapsburg or French) were widely copied, even among people of quite modest means: no respectable housewife would ever dream of not having at least one 'salon' a month, where coffee, cake, and deli-like food was eaten to the accompaniment of a great deal of gossip, cultivated talk, and (some) political discussion. Lavish dinner parties for couples, day trips to surrounding farms and vinyards and intimate, family-friendly (or grown-up after-theater) suppers also were very popular -- it seemed like almost every time of day was fair game to invite a few friends over to eat and chat.
Since everyone aspired to being thought sophisticated, or at least "cultured", dancing, or music alone, was considered an important part of the entertainment, or even as part of a family night spent at home, both in itself and for its educational value, and since a great many more people had had lessons than had money to hire a band, there was a huge demand for music that was playable with the resources available to an average family, say, a husband-and-wife team with a piano. Accordingly, there was a lot of music written in the German-speaking 19th century for piano and one voice, composed by such luminaries such as Franz Schubert, Robert Schumann, Johannes Brahms, and Hugo Wolf, on to Gustav Mahler (and his wife, Alma), and the tradition carries on into the present day with Spanish, American, and British poetry set to music as well.
Although many have pointed out the similarities between lieder and much classic rock and singer-songwriter pop, (The Beatles are often invoked here, though "Love Song" by The Cure better captures the emotional mood, Patti Smith Group or The Doors the wedding of poetry to melody, Sting the seriousness...perhaps the closest match I can think of is "Every Shining Time" by Sunny Day Real Estate) lieder remains one of the most recherche genres of classical music. I cannot imagine why this is so.