Comparison according to Chessopedia

Bishops:

The bishop's predecessor in shatranj (medieval chess) was the alfil, meaning elephant, which could leap two squares along any diagonal, and could jump over an intervening piece. As a consequence, each alfil was restricted to eight squares, and no alfil could attack another. The modern bishop first appeared shortly after 1200 in Courier chess. A piece with this move, called a cocatriz or crocodile, is part of the Grande Acedrex in the game book compiled in 1283 for King Alfonso X of Castile. The game is attributed to "India", then a very vague term. About half a century later Muḥammad ibn Maḥmud al-Āmulī, in his Treasury of the Sciences, describes an expanded form of chess with two pieces moving "like the rook but obliquely."

Derivatives of alfil survive in the languages of the two Countries where chess were first introduced within Western Europe - Italian (alfiere) and Spanish (alfil). It was known as the aufin in French, or the aufin, alphin, or archer in early English.

The term "bishop" first entered the English language in the 16th century, with the first known written example dating back to 1560s. In all other Germanic languages, except for Icelandic, it is called various names, all of which directly translate to English as "runner" or "messenger" (e.g. in Norwegian "Løper", in Danish "Løber", in Swedish "Löpare", in German "Läufer" and in Dutch "loper".) (In Finnish, the word is "lähetti", with the same meaning.) In Romanian, it is known as "nebun" which refers to a crazy person (similarly to the French name "Fou" (fool) which is most likely derived from "Fou du roi", a jester). In Icelandic, however, it is called "biskup", with the same meaning as in English. Interestingly, the use of the term in Icelandic predates that of the English language, as the first mentioning of "biskup" in Icelandic texts dates back to the early part of the 14th century, while the 12th-century Lewis Chessmen portray the bishop as an unambiguously ecclesiastical figure. In The Saga of Earl Mágus, which was written in Iceland somewhere between 1300–1325, it is described how an emperor was checkmated by a bishop. This has led to some speculations as to the origin of the English use of the term "bishop".

The canonical chessmen date back to the Staunton chess set of 1849. The piece's deep groove symbolizes a bishop's (or abbot's) mitre. Some have written that the groove originated from the original form of the piece, an elephant with the groove representing the elephant's tusks (see photo in the history section). The British chose to call the piece a bishop because the projections at the top resembled a mitre. This groove was interpreted differently in different countries as the game moved to Europe; in France, for example, the groove was taken to be a jester's cap, hence in France the bishop is called "fou" (the jester; the word can also mean madman or gannet).

In some Slavic languages (e.g. Czech/Slovak) the bishop is called "střelec/strelec", which directly translates to English as a "shooter" meaning an archer, while in others it is still known as "elephant" (e. g. Russian slon, Turkish fil). In South Slavic languages it is usually known as "lovac", meaning "hunter", or "laufer", taken from the German name for the same piece.[citation needed] An alternative name for bishop in Russian is officer (Russian: офицер).

Mobility:

In general bishops are approximately equal in strength to knights, but depending on the game situation either may have a distinct advantage.

Less experienced players tend to underrate the bishop compared to the knight because the knight can reach all squares and is more adept at forking. More experienced players understand the power of the bishop.

Bishops usually gain in relative strength towards the endgame as more pieces are captured and more open lines become available on which they can operate. A bishop can easily influence both wings simultaneously, whereas a knight is less capable of doing so. In an open endgame, a pair of bishops is decidedly superior to either a bishop and a knight, or two knights. A player possessing a pair of bishops has a strategic weapon in the form of a long-term threat to trade down to an advantageous endgame.

In certain positions a bishop can by itself lose a move (see triangulation and tempo), while a knight can never do so. The bishop is capable of skewering or pinning a piece, while the knight can do neither. A bishop can in some situations hinder a knight from moving. In these situations, the bishop is said to be "dominating" the knight.

On the other hand, in the opening and middle game a bishop may be hemmed in by pawns of both players, and thus be inferior to a knight which can hop over them. Furthermore, on a crowded board a knight has many tactical opportunities to fork two enemy pieces. A bishop can fork, but opportunities are more rare. One such example occurs in the position at right, which arises from the Ruy Lopez: 1.e4 e5 2.Nf3 Nc6 3.Bb5 a6 4.Ba4 Nf6 5.0-0 b5 6.Bb3 Be7?! 7.d4 d6 8.c3 Bg4 9.h3!? Bxf3 10.Qxf3 exd4 11.Qg3 g6 12.Bh6!

A bishop dominating a knight.
[fen]8/8/8/n2B4/8/8/8/8[/fen]

Knights:

A knight is approximately equal in strength and value to a bishop. The bishop has longer range, but is restricted to only half the squares on the board. Since the knight can jump over pieces which obstruct other pieces, it is usually more valuable when the board is more crowded (closed positions). A knight is best when it has a 'support point' or outpost – a relatively sheltered square where it can be positioned to exert its strength remotely. On the fourth rank a knight is comparable in power to a bishop, and on the fifth it is often superior to the bishop, and on the sixth rank it can be a decisive advantage. This is assuming the knight is taking part in the action; a knight on the sixth rank which is not doing anything useful is not a well-placed piece.

A powerful knight occupying a hole (d5) in the enemy pawn structure.
[fen]r2r2k1/pp3pbp/3p2p1/3Np3/4P2q/P4P2/1P1Q2PP/1K1R3R[/fen]

Enemy pawns are very effective at harassing knights because a pawn attacking a knight is not itself attacked by the knight. For this reason, a knight is most effective when placed in a weakness in the opponent's pawn structure, i.e. a square which cannot be attacked by enemy pawns. In the diagram at, White's knight on d5 is very powerful – more powerful than Black's bishop on g7.

Mobility:

Whereas two bishops cover each other's weaknesses, two knights tend not to cooperate with each other as efficiently. As such, a pair of bishops is usually considered better than a pair of knights. World Champion José Raúl Capablanca considered that a queen and a knight is usually a better combination than a queen and a bishop. However, Glenn Flear found no game of Capablanca's that supported his statement and statistics do not support the statement either. In an endgame without other pieces or pawns, two knights generally have a better chance against a queen than two bishops or a bishop and a knight would.

Compared to a bishop, a knight is often not as good in an endgame. The knight's potential range of movement is more limited, which often makes it less suitable in endgames with pawns on both sides of the board. However, this limitation is less important in endgames with pawns on only one side of the board. Knights are superior to bishops in an endgame if all the pawns are on one side of the board. Furthermore, knights have the advantage of being able to control squares of either color, unlike a lone bishop. Nonetheless, a disadvantage of the knight (compared to the other pieces) is that by itself it cannot lose a move to put the opponent in zugzwang (see triangulation and tempo), while a bishop can. In this position, if the knight is on a white square and it is White's turn to move, White cannot win. Similarly, if the knight was on a black square and it was Black's turn to move, White cannot win. In the other two cases, White would win. If instead of the knight, White had a bishop on either color of square, White would win with either side to move.

Knight trapped by an enemy bishop, knight trapped by a king
[fen]8/8/8/8/2B5/8/6K1/2n4n[/fen]

At the end of the game, if one side has only a king and a knight while the other side has only a king, the game is a draw since a checkmate is impossible. When a bare king faces a king and two knights, checkmate can occur only if the opponent commits a blunder by moving his king to a square where it may be checkmated on the next move. Otherwise, a checkmate can never be forced. However checkmate can be forced with a bishop and knight, or with two bishops, even though the bishop and knight are in general about equal in value. Paradoxically, checkmate with two knights sometimes can be forced if the weaker side has a single extra pawn, but this is a curiosity of little practical value (see two knights endgame). Pawnless endings are a rarity, and if the stronger side has even a single pawn, an extra knight should give him an easy win. A bishop can trap (although it cannot then capture) a knight on the rim (diagram), especially in the endgame.