The usual cartridge has resistor elements under the jet holes. When the resistor is heated with electric current, the ink is heated and forced out the hole. These resistors are usually immersed in ink, but if the ink runs out, the resistor is in air and will overheat and can be destroyed. There is another means of forcing the ink out the jet hole which uses crystal elements which bend when a voltage is applied. The bending of the crystal can be used to push out the ink. Link to Information About Piezoelectric Printers
If the cartridge does not print properly, and normal cleaning does not correct the problem, it is now possible to buy kits (about $5) which will clean the jets. The cartridges can be refilled, but can be messy if care is not taken. Use disposable gloves, have lots of newspaper on the area, adhesive tape, scissors, paper tissues and any other items recommended.
With some three colour cartridges inks can mix behind the jets. If one colour contains more ink it can bleed into another colour. This can usually be corrected by printing areas of the colour suspected of being contaminated, or sucking some ink from those jetholes using small diameter plastic tubing. This can also be done to remove bubbles. Determine which colour is being contaminated and add more ink to that container. The leakage can be slow, so check often after refilling. Keep the quantity of added ink small, perhaps 1 or 2 cc and do not push the needle more than half way down the sponge and gently add ink until it appears at the top.
Link to Refilling Instructions
People have been refilling black ink cartridges with good grade ink bought from a stationary shop. Link to Using Ordinary Black Ink - Find Ink Refilling
Sometimes a vacuum is used to hold the ink in the container. The smallest air leak will cause ink leakage. Plain water has been used with satisfactory results if there is a foam pad in the cartridge. It should not cause any harm. If the printer has a plugin cartridge and is not used for a long time, it is worthwhile to remove the cartridge, put it in a small open container and place the container, along with a wad of paper saturated with water into a roomy plastic bag and sealing it securely. Test often with a dab of the jet holes onto a tissue.

Photos of the business end of a HP inkjet cartridge
The plate with the jet holes has been removed

More photos taken with a video camera
through a microscope
There was nothing which looked
like heating resistors.
Along the edge of the opening
into the interior of the cartridge there
was a strip of bright metal, whereas the
rest of the surface was coated with varnish.
The connecting strips ended with a small
semicircular raised section which could
have been the position of the jet.

An idea for an easy to refill ink cartridge