Basically there are two kinds of spinning wheels, the wool wheel (great wheel or walking wheel) and a flax wheel. The wool wheel would spin other fibers besides wool such as cotton, animal hair, etc. Whereas the flax wheel (smaller than the wool wheel) spun flax to make linen. The flax wheel would also double as the distaff.
The spinning wheel was not invented in the 19th century in fact it dates back to 13th century but they were still commonly used during the 19th century.
Today they hold a romantic interest but amazingly they did during the 19th century as well, you can find novels, songs, poems and stories from that time period where the spinning wheel was central to the story. Here's a link to Louise May Alcott's "Spinning-wheel stories," as a sample of some of the work of the period. Even Longfellow wrote a poem "The Spinning-Wheel."
As factories sprang up across the United States, fewer and fewer people were spinning their own threads to make their own cloth. The markets were bringing in cloth, pre-made, pre-patterned and the trend from making your own shifted to buying more pre-made cloth. By the end of the century there were still women weaving but it was quickly becoming a dying art.
In the past 30 years there's been a renewed interest in spinning but more as a hobby not on a need to provide cloth and clothing for the family.