With cellular radio, a simple hexagon is used to represent a complex object: the geographical area covered by cellular radio antennas. These areas are called “cells.” This shape lets us picture the cellular idea on a map, because when displaying a cellular system we want to depict an area totally covered by radio—without any gaps. Any cellular system will have gaps in coverage, but the hexagonal shape lets us neatly visualize, in theory, how the system is laid out.

By definition, a cell site gives radio coverage to a cell. The cell site is a location or a point; the cell is a wide geographical area. The picture illustrates the cell structure. Some people historically saw the cell as the blue hexagon shown, being defined by the cell site in the center and the antenna coverage being the coverage of the hexagon surrounding it. This held true for historical omni-directional antenna systems. As new directional antenna systems have been deployed, the cell site and the cell required clearer definition, as more sectors or cells are associated to a cell site.

In reality in today’s systems, the cells are the red hexagons, with the cell sites or base stations at the corners. Rather than referring to a “three-sectored cell,” it is more appropriate to refer to the three “cells” associated with a single base station, and as it happens, each is a sector. In addition, rather than those three cells being enclosed in a single hexagon, each cell is represented by its own hexagon. So in effect, sectors and cells are the same thing in a three-sector situation
https://www.commscope.com/uploadedImages/CommScope.com/Blog/...
https://www.commscope.com/Blog/Cells--Sectors-and-Antenna-Be...
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As the phone user moves from one cell area to another cell while a call is in progress, the mobile station will search for a new channel to attach to in order not to drop the call. Once a new channel is found, the network will command the mobile unit to switch to the new channel and at the same time switch the call onto the new channel.
https://en.wikipedia.org/wiki/Cellular_network