The filament is the source of electrons (cathode) in x-ray tubes. A thin wire (0.1- 0.5 mm, usually tungsten) emits electrons due to thermionic emission, operating in a vacuum and energized with electric current.
A CT tube utilizes a larger filament with larger size of the effective focal spot than a conventional x-ray tube.

The focal spot is the point where the electron beam impinges on the tube anode and from which x-rays are emitted. The size of the focal spot is determined by the size of the filament and cathode, which is determined by the manufacturer.
Most x-ray tubes have more than one focal spot size. The size of the focal spot has influence on spatial resolution. The smaller the focal spot, the better the limiting spatial resolution of the x-ray system, especially in magnification mammography. The use of a small focal spot concentrates heat onto a smaller portion of the anode therefore, more heat is generated and a longer cooling time is necessary.

X-ray tubes are devices for the production of x-rays. X-ray tubes consist of an evacuated glass vessel and two electrodes. An electrical current with very high voltage passes across the tube and accelerates electrons emitted by thermionic emission from a tungsten filament (cathode also called electron gun) towards the anode target. The electrons collide with the anode and this deceleration generates x-rays (bremsstrahlung).
The high vacuum allows the electron beam an unimpeded passage. The electron beam heats the anode (usually copper), which is cooled by water to prevent melting. A copper target emits x-rays with a characteristic wavelength. Other used metals soften or harden the x-ray beam.
The x-rays pass through a very thin beryllium (Be) foil. This beryllium window absorbs a high amount of the elastically scattered electrons (produced by the target) and allows the radiation to get out of the tube without substantial absorption.
In conventional x-ray tubes, the anode is also the target. In nanofocus and microfocus x-ray tubes, the electron beam is transmitted through a hole in the anode where it is then focused onto a small spot on the target.

See also X-Ray Tube Housing, Fine Focus X-Ray Tube, Transformer, Diode, Digital to Analog Converter and Angular Response.