Dvořák himself was reluctant to compose a concerto for cello for a long time because of concerns about its viability as a solo instrument against a full orchestra. Bear in mind that all the previous concertos for solo cello were for fairly small orchestra, not the large romantic orchestra that Dvořák wrote for. He may have finally been convinced by hearing a couple of performances of a cello concerto written by Victor Herbert in New York where Dvořák was directing the National Conservatory. In any case, he took up the task and delivered the first great cello concerto that established the genre and prompted many other composers in the 20th century to write concertos for the cello.
The piece is substantial, about the same length as the Beethoven Violin Concerto. It follows the usual fast-slow-fast three movement form. Dvořák follows the example of Beethoven and Brahms in giving a full orchestral tutti before the soloist enters. The orchestra gets to present all the main themes, which the cello then re-presents. If not well-written, this might feel redundant, but the cello part is so resonant and richly ornamented that it is not a problem. Indeed, you might start to think that the typical romantic ploy of giving a big bombastic opening to the solo instrument as we have seen in Grieg, Tchaikovsky and others, is just a bit gimmicky.
Here is the first theme, vaguely folk-like sounding because of the lowered 7th degree, given to the clarinet in A:
And here is how the cello presents that same theme (bass clef):
Here is a lyric theme first given to the horns:
And here is how the cello handles that theme (beginning in bass clef):
I won't go picking apart the rest of this lovely concerto. As is usual with Dvořák the themes are song-like and memorable. There is an inherent lyricism in this concerto, even though the cello part is very virtuosic with some very high registers and some difficult double-stopping.
Here is a great performance by Jacqueline du Pré, the uniquely talented cellist whose career ended tragically young, at age 28, when she contracted multiple sclerosis.