A driver, short for “device driver”, is software installed into Windows (or any operating system) that handles the operating details for using a particular hardware device.
For example, a mouse can be connected to your computer any of several ways: via a wired PS2 connector, a wired USB connector, a wireless USB connector, or a Bluetooth. Each hardware interface has a unique way of getting the information from the mouse. The mouse driver for each different interface translates that hardware-specific information into generic information, like “the right mouse button is pressed”, that Windows then uses.
Many drivers are included with Windows, and these can be overwritten by drivers that you install or download from device manufacturers. This allows Windows to take advantage of more of the hardware capabilities of the specific devices. For example, Windows includes a fairly generic mouse driver, but downloading updated drivers from the mouse manufacturer may enable additional buttons and capabilities that the default drivers ignore.
Drivers can be simple or complex, and like any software, drivers can have bugs. Quite often the advice when diagnosing suspected hardware-related issues is to download the latest drivers directly from the hardware manufacturer, in the hopes that bugs present in earlier versions, including the default drivers included with Windows, have been fixed.