Technically speaking, it's not really ink — it's actually pigment (generally metal salts or even plastics) suspended in a carrier solution which keeps the pigments evenly mixed, applicable, and clean.
Some tattoo artists mix their own ink, although most buy it pre-mixed. The advantage to making your own is that you're more likely to know exactly what it is, and you can mix a carrier that suits your needs (for example, ensuring that it is vegan friendly). A typical "home made" carrier solution might be a mix of a small bottle full of vodka (or listerine, etc.) with a little glycerine and propylene glycol (if too much of these are used, the ink will be oily and may separate), mixed into a slurry with the dry pigments.
Other common carrier components both in home made and commercial tattoo inks include alcohols, anti-freeze (ack! toxic!), formaldehyde (very toxic!), gluteraldehyde (also toxic!), water, witch hazel, and more—although a common professional base ingredient list might read "sterilized water, alcohol, glycerine". Ask your artist—if they don't know what's in their tattoo ink, do you really want to be tattooed by them?
The following is a rundown as to the common ingredients of the different colors:
* BLACK: made of iron oxides, carbon, or logwood. "Natural black pigment is made from magnetite crystals, powdered jet, wustite, bone black,and amorphous carbon from combustion (soot). Black pigment is commonly made into India ink. Logwood is a heartwood extract from Haematoxylon campechisnum, found in Central America and the West Indies."
* BROWNS, FLESHTONES: made of ochre. "Ochre is composed of iron (ferric) oxides mixed with clay. Raw ochre is yellowish. When dehydrated through heating, ochre changes to a reddish color."
* RED: made of cinnabar, cadmium red, iron oxide, or napthol. "Iron oxide is also known as common rust. Cinnabar and cadmium pigments are highly toxic. Napthol reds are synthesized from Naptha. Fewer reactions have been reported with naphthol red than the other pigments, but all reds carry risks of allergic or other reactions."
* ORANGE: made of disazodiarylide, disazopyrazolone, or cadmium seleno-sulfide. "The organics are formed from the condensation of 2 monoazo pigment molecules. They are large molecules with good thermal stability and colorfastness."
* YELLOW: made of cadmium yellow, ochres, curcuma yellow, chrome yellow, or disazodiarylide. "Curcuma is derived from plants of the ginger family; aka tumeric or curcurmin. Reactions are commonly associated with yellow pigments, in part because more pigment is needed to achieve a bright color."
* GREEN: made of chromium oxide ("Casalis Green" or "Anadomis Green"), Malachite, Ferrocyanides, Ferricyanides, Lead chromate, Monoazo pigment, Cu/Al phthalocyanine, or Cu phthalocyanine. "The greens often include admixtures, such as potassium ferrocyanide (yellow or red) and ferric ferrocyanide (Prussian Blue)."
* BLUE: made of azure blue, cobalt blue, or Cu-phtalocyanine. "Blue pigments from minerals include copper (II) carbonate (azurite), sodium aluminum silicate (lapis lazuli), calcium copper silicate (Egyptian Blue), other cobalt aluminum oxides and chromium oxides. The safest blues and greens are copper salts, such as copper pthalocyanine. Copper pthalocyanine pigments have FDA approval for use in infant furniture and toys and contact lenses. The copper-based pigments are considerably safer or more stable than cobalt or ultramarine pigments."
* VIOLET: made of manganese violet (manganese ammonium pyrophosphate), quinacridone, dioxazine/carbazole, and various aluminum salts. "Some of the purples, especially the bright magentas, are photoreactive and lose their color after prolonged exposure to light. Dioxazine and carbazole result in the most stable purple pigments."
* WHITE: made of lead white (lead carbonate), titanium dioxide, barium sulfate, or zinc oxide. "Some white pigments are derived from anatase or rutile. White pigment may be used alone or to dilute the intensity of other pigments. Titanium oxides are one of the least reactive white pigments."
It should be noted that some inks can be harmed by direct sunlight—so inks should not be stored in clear bottles on the window sill! Heat can also alter some inks, so ink can not be sterilized in a heat-based autoclave.
Tattoo inks are unregulated and not "FDA approved" or any such thing. Some of the ingredients used in tattoo ink are approved for use in cosmetics, foods, and medical devices (including iron oxides, logwood, and titanium dioxide), although most are not. However, ink allergy is extremely rare—after all, somewhere in the range of five million tattoos are done yearly with almost no complications of this type. That said, if they do occur, see a dermatologist experienced in dealing with tattoos.
Hope this helps...