Chemical properties
Tannic Acid (Acidum Tannicum) is, as the name suggests, a Tannin. It is an ester of gallic acid (3,4,5-hydroxybenzoic acid) and glucose, that is, galloyl tannin (thank you vuo) and has the chemical formula C14H10O9 and is derived from Oak. Tannic Acid is somewhat of a misnomer, as this chemical is actually not an acid as such, but a polyphenol compound. It is soluble in water (2 in 1), alcohol (10 in 6) and slowly, in glycerin (1 in 3, or 1 in 1 on heating). It is, however, almost insoluble in pure ether, chloroform, or benzene.

Use in tanning
It is one of many ways of tanning animal hide, thus producing leather. It is classified as a Vegetable tannin, which works by transforming the gelatin in the animal hide into a water insoluble form and thereby making the skin somewhat water-resistant. It was for this reason that tannins first became tools for humans, but they do have other uses, notable in the staining of woods.

Medical uses
When taken orally, tannic acid has an astringent property, that is it binds tissues. It will penetrate (although not deeply) into cells in the mouth, and cause a general feeling of astringency, in the stomach it forms coagulated proteins and in the gut tends to constipate. For this reason, it is sometimes prescribed to stop diarrhoea. It is also used externally on wounds, to act as a coagulant. It is rarely taken internally, due to these coagulating properties. A typical dose is 1-3 decigrammes.

Warning: Tannic acid is toxic. Do not ingest it unless a doctor has prescribed it to you. Regular exposure can lead to liver or kidney damage, any exposure can result in irritation of the area exposed, or a binding and constipated intestinal tract for ingestion. Inhalation can induce sneezing and a general bound feeling. If it is exposed to the eyes, or a rash develops from skin contact, seek medical attention immediately. If it is ingested vomiting should be induced, but it would be best to consult somebody with medical training for this.

Use in wood staining
It is also used in the staining of woods, as many wood stains react with tannic acid woods with a low tannin content need to be first treated with tannic acid, in order to ensure they can be stained. The mixture required to stain the wood is 1 part tannin to 19 parts water. The staining itself is often done with Potassium Dichromate or Ammonia to produce wood tones.

Solubility and medical applications from: http://www.ibiblio.org/herbmed/eclectic/bpc1911/acidum-tann.html
Wood staining information from: http://www.woodfinishsupply.com/Tannin.html
Tannin
Tanning
Thanks to vuo for a correction about the compound nature of this chemical and excellent description of the chemical.
vuo used: http://medplant.nmsu.edu/Frhus.html