It's a rock, smoothed and rounded on one side. They were laid as pavers, predating tarmac, poured concrete, and asphalt. They can be set in sand or dirt, or held together with mortar, but the granite cobblestones on the Memphis riverfront don't have any mortar, so they're flexible, resistant to cracking, and permeable.
You can buy cobblestones today, old and new ones. They've become popular for paving newly pedestrianized streets and high quality drives and walkways. Here's a video on tips for installing. The narrator's sense of humor and Irish accent reminds us that it was Irish immigrants who brought their masonry skills to Memphis and laid the stones on our landing.
Laying Cobble Stone Tips - The best video clips are here