A knife is a shorter version of a sword. It has a short blade and a small hilt.
Knives are good for close-range combat. While not having the same reach or power as swords, their smaller size allows them to easily exploit openings (e.g. chinks in armor) and quick strikes. Demigod knives are either made out of Celestial Bronze or Imperial Gold, so they can vaporize monsters or be used on other demigods or gods. Celestial Bronze knives are very dangerous to demigods and monsters alike. According to Luke Castellan, you have to be clever and fast and very skilled to fight with a knife and he feels that Annabeth Chase is clever enough to use one. She has fought with the knife that he gave her ever since.
Like Artemis, the Hunters of Artemis also utilize their knives made of silver during close combat. Each of the knives are long as a persons' arm.
The prophecy in The Last Olympian says "The hero's soul, cursed blade shall reap," and Percy thought it would either be Riptide or Kronos' scythe, but it turned out to be Annabeth's Knife because Luke broke his promise to Annabeth when he and Thalia first met her.
Helen of Troy owned a knife she called Katoptris, meaning looking glass, because she only ever used it as a mirror, never for combat. Such a knife, according to Annabeth, is more ceremonial. Piper McLean, a daughter of Aphrodite, recently obtained this very knife and can use it competently in a fight. Piper uses charmspeak to cause her opponents to let down their guards and uses her knife to dispatch opponents at her leisure.