Wikipedia defines the horizon effect as: a problem in artificial intelligence whereby, in many games, the number of possible states or positions is immense and computers can only feasibly search a small portion of them, typically a few plies down the game tree. Thus, for a computer searching only five plies, there is a possibility that it will make a detrimental move, but the effect is not visible because the computer does not search to the depth of the error (i.e., beyond its “horizon”).
What it means, in more understandable words, is that when a chess computer finds a move, or a series of moves, that loses material, or some other advantage, it stops analyzing that move or series of moves. This can lose the game, or at least the advantage, as it fails to see a strong reply or the continuation of play that will allow it to retain or increase its advantage.
An early example of the horizon effect can be found in this game.
De Legal-Saint Brie?
1.e4 e5 2.Nf3 d6 3.Bc4 Bg4 4.Nc3 Nc6
5.Nxe5 Bxd1?? (There were many computers in the early 1980’s would simply take the offered queen, as it was taught that being up a queen would lead to victory and would therefore stop analyzing. This simple trap caused consternation and scorn by some players as they wanted a “serious” chess computer. By the way, this trap is known as De Legal’s mate.) 6.Bxf7+ Ke7 7.Nd5mate 1-0
A more recent example can be found in this game:
Najdorf Thematic Tournment
Chess.com, July-August 2019
1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 a6 6.Bc4 g6 7.f3 Bg7 8.Be3 O-O 9.Qd2 Nc6 10.O-O-O Bd7 (The Najdorf has transposed into a Dragon, B77 to be exact.) 11.g4 Rc8 12.Be2 Ne5 13.h4 Nc4 14.Bxc4 Rxc4 15.h5 Qc7 16.Kb1 Rc8 17.hxg6 hxg6 18.Nde2 (This is an important move as it provides another piece to guard c3 and puts a stop to Black’s attack.) 18…Be6 19.Bh6 Bh8?
20.Bf8! (This keeps the Black’s king from escaping to the center.) 20…Kxf8 (Not 20…Rxf8 21.Qh6! +-. Best for Black is 20…Nh5 21.Rxh5 gxh5 22.Qh6 Rxf8 23.Rh1 Bg7 24.Qxh5, and now if 24…f5 25.Nf4! wins on the spot.) 21.Rxh8+ Ng8
22.Rxg8+! (The chess.com computer recommends 22.Qh6+ Ke8 23.Rxg8+ Kd7 24.Rxc8 Qxc8 25.e5 Kc7 26.exd6+ exd6, when White is obviously winning. But the text move is better as it leads to a forced mate. So why did chess.com computer miss this move? Probably because it saw that White loses the exchange and concluded that’s not a good way to proceed. So it stopped analyzing.) 22…Kxg8 23.Qh6! f6 (Black is in Zugzwang, as his king is paralyzed and he can’t get help in time. 23…d5 24.Rh1 +-) 24.Qxg6+ Kf8 25.Rh1 1-0 (25…Bg8 26.Rh8 e6 and now either 27.Qxg8+ or 27.Rxg8+ mates.)