The most important function of a scoreboard is that it be visible and easily readable to both fans and players alike. This requires that it be fairly large in size and mountable or holdable in such a position that it can be seen from a solid distance. For best results, The numbers of the actual score should be, at minimum, a foot tall and at least six inches wide. The numbers should also be in a color that contrasts clearly with their background. White on a black background is a popular scheme, as well as neon colors for numbers on a black background.
Your scoreboard should also be mounted high or tall enough to be seen across the field. Mounting it so the numbers are about eye level is a good, rough-estimate measurement for placing it high enough.
In some sports, such as basketball, the scores change rapidly all game long. Having a scoreboard that is easily accessible as well as easy to change the score is necessary. Many simple scoreboards use simple rotating number placards made from wood or heavy plastic. Having two for each team allows you to easily score from 00 to 99. Some companies make high-visibility numerical modules that change numbers by flipping switches or levers.
If you're designing your scoreboard for use in a specific sport, you'll also need to plan on having room for sport-specific details. For instance, knowing when each team is in "bonus" for basketball or what half of what inning it is for baseball are important for both fans and players to know. However, they aren't necessarily as important as the score and can be significantly smaller. This also helps avoid confusion as to which numbers are the score and which aren't.