As Rnceus.com explains, the liver has a critical role in the processing of toxins, including some medications (such as acetaminophen) and alcohol. The liver contains many enzymes, which are special proteins that can enable chemical reactions to take place in the body. Some of the enzymes in the liver are used for toxin processing. Blood from the body flows through the liver where toxins in the blood can be processed. Some of the enzymes work to inactivate toxins to keep them from damaging the body. Enzymes can also break down compounds in the blood or modify them so that they stay dissolved and can then be excreted, via the kidneys or other methods.
According to the Nuffeld Department of Anethesiology (which is associated with the Oxford Medical Center in England), the kidneys are one of the primary ways in which the body is able to eliminate toxins and waste products from the blood. The kidney contains many small structures called glomeruli, which essentially work to filter substances (such as toxins and waste products) out of the blood, where they are concentrated into the urine, which is then excreted from the body via the bladder and the urethra. Some toxins don't get filtered out by the glomeruli and are instead actively secreted by the kidneys into the urine. Regardless, the kidneys ability to secrete waste products and toxins into the urine is one of the primary ways in which toxins are eliminated from the body.
Another way in which toxins can be eliminated from the body is via sweat. Some toxins and waste products in the blood are able to diffuse into the sweat glands. As a result, when the body excretes sweat (in order to cool down), some toxins are excreted as well. Sweating is normally not able to process nearly as many toxins as urine production by the kidneys, but it provides an auxiliary method of toxin elimination.