A scoreboard started as a person with a pile of numbers. When a point was scored, somebody put the matching digits on a hook. Almost every major league facility now has a video board as its scoreboard or a matrix display, and those boards are usually driven by programs that keep statistics, not just the score.
From light bulbs to LEDs
Until the 1980s, electronic scoreboards were mostly electro-mechanical, with relays or stepping switches lighting digits made of incandescent bulbs. From the 1980s, solid-state electronics allowed major improvements. Light-emitting diodes first replaced bulbs on indoor scoreboards and, as they got brighter, on outdoor ones. LEDs last many times as long as bulbs, do not break, and turn electricity into light much more efficiently.
The jumbotron
The large video screen was invented in Japan in the early 1980s, and two companies dispute who did it first. In 1980 Mitsubishi Electric introduced the first large-scale video board, called Diamond Vision, which used cathode-ray tube technology. Its first public showing was at Dodger Stadium in Los Angeles, during the 1980 baseball All-Star Game. Sony registered JumboTron as a trademark in 1985. The first jumbotrons used cathode-ray tubes in modules, and the technology was later replaced by LEDs. Today the word is used for any large video wall in a stadium or concert venue.
Who supplies them
Daktronics, a company based in Brookings, South Dakota, designs and manufactures video displays, scoreboards, digital billboards and sound systems. It was started in 1968 by a pair of professors from South Dakota State University. At State Farm Stadium, for example, Daktronics LED video and ribbon displays were installed in 2006, before the Arizona Cardinals' first game there.
The table below only counts suppliers that a venue's Wikipedia infobox names. Most venues' articles say only that a scoreboard exists, so the real picture is wider than the list.