In a clinical context:

Dialysis is a method for removing waste such as urea from the blood when the kidneys can no longer do the job. The two types of dialysis are: hemodialysis and peritoneal dialysis.

In hemodialysis, the patient's blood is passed through a tube into a machine that filters out waste products. The cleansed blood is then returned to the body.

In peritoneal dialysis, a special solution is run through a tube into the peritoneum, a thin tissue that lines the cavity of the abdomen. The body's waste products are removed through the tube.

There are three types of peritoneal dialysis. Continuous ambulatory peritoneal dialysis (CAPD), the most common type, needs no machine and can be done at home. Continuous cyclic peritoneal dialysis (CCPD) uses a machine and is usually performed at night when the person is sleeping. Intermittent peritoneal dialysis (IPD) uses the same type of machine as CCPD, but is usually done in the hospital because treatment takes longer. Hemodialysis and peritoneal dialysis may be used to treat people with diabetes who have kidney failure.

It works by having the blood flow along one side of a semi-permeable membrane, with the dialysis solution (usually a highly concentrated saline) flowing along the other side. Due to the difference in osmolarity between the two liquids, water traverses the membrane in order to dilute the dialysis liquid, carrying along the unwanted blood contents.

Original version from the Diabetes dictionary

In a biochemical laboratory context:

Dialysis also refers to a common laboratory technique which operates on the same principle as clinical dialysis described above. Typically a solution of several types of molecules is placed in dialysis tubing, and the tubing is sealed. The sealed tube is placed jar of a different solution, or pure water. Molecules small enough to pass through the tubing (often water, salts and other small molecules) tend to move into or out of the tubing, in the direction of decreasing concentration. Larger molecules (often proteins, DNA, or polysaccharides) cannot pass through the tubing. One common reason for using this technique would be to remove the salt from a protein solution.