Familiars (使い魔, tsukaima?) are special assistants used by the magi. Since it's natural for a magus to separate himself from society, the use of familiars capable of performing small, simple errands is considered orthodox. Although a different entity, familiars are seen as an extension of the magus as there is both a mental and physical connection between the two. Fundamentally different from creatures that are summoned and then bound by a contract.
Nature
There are several ways to create a familiar. The most traditional is by simply transferring a part of the magus body that has a great concentration of Magic Circuits (eyes, hair, blood, etc.) into the body of a dead creature in order to give it a new life. The resulting familiar will gain Magic Circuits that the original never had before and will be able to use Magecraft just like the magus. However, prana cannot be produced independently and must be supplied by the magus through their spiritual connection. Thus, a familiar requires a master to maintain both his powers and life. On a side note, modern versions of this traditional type are usually very weak.
A more powerful variant of the traditional would involve adding the soul of a dead person into the corpse during the creation of the familiar. The familiar will then form its own personality, being capable of reasoning and acquiring experience apart from the magus while sharing the same knowledge. There is also a chance of the arising of abnormal powers. A good example of this would be Len, who not only has succubus-like aspects but is also capable of living for long periods of time without a master and forging contracts with individuals other than her creator. However, the presence of a personality also makes this type of familiar more independent from the magus. Important to note that this is by no means a form of resurrection, as the familiar is still a being completely different from the one whose soul it received.
Additional notes
Familiars like the ones mentioned above can be made out of pretty much any creature: cats, dogs, birds, etc. Humans can also be turned into familiars, but that is seen as a monstrous act and unpractical (due to prana burden) by most magi.
Completely different types of familiars also exist. Examples of those include Cornelius Alba's slimes, magic constructs such as Aozaki Tohko's puppets, and spiritual beings such as human ghosts and nature spirits.
Note that, strictly speaking, a Servant is not a familiar and rather falls more towards the side of summoned being. Servants are called familiars only due to some similarities (spiritual connection, dependence on prana) and for the sake of formalities. They need to be put on a class of their own however, as it would be impossible to summon Heroic Spirits without the help of the Great Grail and there is no need to use something as powerful as Command Spells to control regular familiars.
Transfer of the master's consciousness into the body of regular familiars is possible and most magi do so to remote control them while they are in recon or exploring missions. Yet, the transfer is not complete and if the magus' body dies, his soul will die with it. It is impossible to perform transfer of consciousness with a familiar with its own soul and personality. In this case, the best that the magus can hope for is sharing of senses with familiar (like receiving a video-feed of everything that the familiar sees and hears).
List of Familiars
Black Hound
黒い猟犬 - Kuroi ryouken
This black Doberman is Cornelius Alba’s favorite familiar. He, who isn’t able to love people, is really fond of animals who could love others easily.
Dragon Tooth Warrior
竜牙兵 - Ryuugahei
Caster's familiars. Created by having fang of dragons scattered over the ground. Final result is a type of particularly strong skeleton footsoldier that receives prana of dragons and knowledge of the earth.
Ghost Liner
ゴーストライナー
The greatest type of familiars. Servants are close to this type.
Owls
梟 - Fukurou
Small statues of amethyst in form of owls that Tohsaka Rin uses for reconnaissance purposes.
Projection Machine
幻燈機械 - Gentou kikai
Created by Aozaki Tohko to replace the familiar that was destroyed by her sister. A suitcase with a projector installed inside. It shapes Ether Clump in the form of a monstrous cat that cannot be destroyed unless the suitcase is destroyed. It also has the ability to heal Tohko by enveloping her.
Puppets
人形 - Ningyou
A special type of familiar that are not any kind of life form, but rather mystic constructs whose actions are controlled (directly or indirectly) by the magus. Their users are known as Puppet Masters (人形遣い, Ningyou tsukai?) and follow the philosophy that the magus himself does not need to be strong, but rather he needs to make the strongest creation. Known Puppet Masters are Aozaki Touko, Cornelius Alba, Van Fem and the unnamed Master of Assassin during the Third Holy Grail War.
Aoko Puppets
青子人形
A pair of puppets that Aozaki Tohko crafted to resemble her little sister, there is one Spider-Model (蜘蛛形態?) and one Crow-Model (鴉形態?). Both can self-destruct.
Automata
自動人形 - Jidou ningyou
Automated puppets shaped after humans that the magus can control as if they were an extension of his limbs. Puppet Masters that use these usually are not suited for direct combat. Examples include the French dolls-type that the Master of Assassin mentioned above used, which were equipped with deadly weapons for precision killing.
Dark Castle
魔城
Giant golems created by Van-Fem, there were seven of them at first, but #5 was destroyed by Fina-blood Svelten.
Slimes
スライム
Created by Cornelius Alba with Ether Clump.
Sealed Box
封印の匣 - Fuuin no hako
Another of Aozaki Tohko’s familiars, a black monster with a thousand mouths and thorn-like tentacles stuck inside a box. It drags its victims to inside the box so it can eat them.
Shadow Giants
影の巨人 - Kage no kyojin
Matou Sakura’s familiars made out of her Shadow element. Under normal circumstances they are extremely small but, when her powers are backed-up by Angra Mainyu and the Holy Grail, their sizes increases vastly and start acting as a materialized curse. Each of them is composed of roughly 1000 units of prana.
The Dead
死者 - Shisha
The deceased. The familiars of the vampires. Created when the corpse of the one who had his blood sucked and then received some of the vampire's own blood does not manage to become a Ghoul. Several ranks stronger than the Living Dead, but still mindless and soulless puppets that are controlled by their creator. They consume meat to maintain their bodies but over half of the absorbed energy is sent to their master.
Worms
淫虫 - Inchuu
The lowest of all familiars. Used by Matou Zouken. Loves feasting on the energy of humans. When the host is a male, they consume their bone marrow and part of the brain after shattering his spine, making him crippled in the process. When it is a female, they absorb the mental energy created after sexual stimulation is made through the nervous system. Usually don't consume the flesh of women, but the uterus is their favorite meals. Will eat the placenta after giving its victim sexual pleasure strong enough to fry the nerves of the brain. Shinji's Book of False Attendent is made out of these.