Basic economic unit of the game.
The Worker is vital for a civilization - absolutely the most important civilian unit in the game. It builds all land improvements like Farms and Mines that give access to resources and improve tile productivity. It also constructs roads and railroads to connect the cities, both in an empire and city-states. Finally, it can repair damaged improvements.
Workers may also clean tiles from vegetation or marshland, thus freeing resources found on them for exploration.
Each of the individual improvement types, as well as the different Remove actions, only become available after you research certain technologies.
To build an improvement, the Worker needs to be within a nation's territory, so you must make sure first to expand your borders to the tile where you want to build. Roads and railroads, however, can be built both within and outside of a nation's territory. Likewise, Forts may be built and the Remove Jungle/Forest/Marsh actions may be performed outside of your territory.
While Workers cannot be gifted to a city-state, they can perform tasks such as repairing pillaged tiles in one.
The Worker is a civilian unit. While it can clear a vacant barbarian encampment, it cannot defend itself, so accompany it with a military unit when it is in dangerous territory. Otherwise you risk him being captured by the enemy!
On the other hand, the Worker can be used as a scout during a war because instead of being damaged or destroyed, it will only be captured and can be easily recaptured at a later time.
Once a player researches Industrialization, all of his or her Workers change in appearance, looking more like stereotypical factory workers.
To survive, a civilization needs skilled workers. They may be highly trained engineers, or civilians conscripted into labor - or, indeed, slaves - but someone has to dig the mines, plant the fields and pave the roads. Without these men and women, a civilization will stagnate and eventually fall prey to a more vigorous neighbor.