Scorebook, scoreboard, scoring record

The scorebook contains the written scoring record of the match, kept by the scorers. In the very early days of the game this record was not 'written' at all - the score was kept by cutting notches into sticks of wood - hence the use of the term notchers for scorers. Some scorers today don't write anything, either - using the ubiquitous laptop computer to record happenings on the field.
There are two popular systems for keeping a written record - the most common of which is the Box system, as used in much club cricket. This is named after Charles Box, author of The Theory and Practice of Cricket, published in 1868. The other is the more informative linear system, invented by Australian W. H. Ferguson in 1905 for the Ashes series in England and subsequently refined by BBC radio scorer Bill Frindall amongst others.

A scoreboard is a device for conveying key information about the state of the match to the spectators. It can be a simple manually-operated device, using painted numbers on metal plates - known as tins - hung on a board, right through to today's ultra-informative electronic extravaganzas that tell onlookers everything except the name of the umpire's dog!