There are 4 basic categories.
FD and C dyes (Food, Drug and Cosmetic Act), pigments, micas and natural colorants.
FD and C colorants are synthetic and are available in a wide range of colors in both powdered and liquid form. In soap making they are mainly used in melt and pour soaps as they are not stable in cold process soap making, often "morphing" (turn into strange colors or disappear altogether). The downside of using these colors in your melt and pour soap is they tend to bleed (migrate from the original site). They are great for making beautiful colors if you are making the bar one color.
Pigments include both ultramarines and oxides. Although many pigments were once taken naturally from the earth, they are now replicated synthetically in a lab for consistency and safety. This was enforced by the FDA because when mined naturally, pigments contain toxic materials which the FDA deemed unsafe.
Ultramarines and oxides are very stable in soap making and can be used in pretty much all soap making processes including melt and pour, cold process and hot process. Pigments are probably the best synthetic color source for swirls as they will not bleed at all. Pigments do tend to clump, so it is best to mix into a small amount of soap first and then add it to the rest.
Micas are colorants which have a glittering effect to them when the proper lighting is present and are available in a wide variety of colors. Mica is a mineral which is used in practically everything you see that is glittery-- plastic, paint, lipstick, nail polish, glitter. It comes in varying size flakes, which determines how sparkly it is. The larger the mica particle, the more light it reflects. Micas are available in a large variety of colors. They are very easy to use in soap making as you don't have to worry about clumping like you do with pigments, however, keep in mind that it does generally take a substantial amount of mica to color a full batch of soap. Mica's are is best seen in soaps made with a clear melt and pour base. Not all mica's are safe for ingestion (use in lip products) so be sure to check with your supplier before using in this manner.
Natural colorants are natural botanicals which you can use to color your soap. Annatto seeds work beautifully for shades of yellow to orange. To use them, soak the seeds in oil for a day or longer, drain the oil off, and use it in your soap. Save the seeds, they're reusable many times! Powdered rosebuds make a nice shade of brown. Red sandalwood powder makes a striking maroon color. Alkanet root powder makes a shade of purple, and madder root powder turns red.
Natural herbs have been used to tint and color products for centuries. These herbs can be used directly in soaps to achieve color or a speckled effect. Alternatively, you can make an oil infusion by warming the herb in oil first. The strained oil can then be used to tint your formulations. To make an infusion, place your herb in a double boiler and then covering it with olive oil. Allow the oil to warm for 2 hours and then check the color. If you want a darker color, remove the herb and replace with new and warm the same oil for another 2 hours.
Here are some other options for natural colorants:
Alkanet - steep in oil first - deep purple to muted blue
Annatto Seed - steep in oil first - yellow to orange
Beet Root - pink to brown
Bentonite Clay - off white to light green
Ground Calendula Petals - yellow
Carrots shredded - yellow to orange
Chamomile - beige to yellow
Cinnamon - tan to brown
Cloves - brown
Cocoa Powder - brown
Coffee Grounds - brown to black
Comfrey Root - tan to light brown
Cucumber - light green
Curry Powder - yellow
Indigo Root - dark blue
Kaolin Clay - off white
Kelp - green
Morrocan Red Clay - dark red
Paprika - Peach to orange to brown (can be an irritant)
Pumpkin - orange
Pink Rose Clay - pink to red
Saffron - yellow
Sage - green
Spinach - green
Tumeric - orange to amber
Wheatgrass Juice - dark green