Dyes are derived from many sources.
If you pick roses and dandelions, and put them in your crafting slot, you get red and yellow dye. Cactus blocks can be cooked in a furnace to create green dye.
Lapis Lazuli, Bone Meal, Ink sac and Cocoa Beans act as dye when applied to sheep or wool, for blue, white, black and brown respectively.
Lastly just like regular paints, some of the above can be combined. Yellow + Red makes Orange Dye for example. Some can be lightened by Bone Meal, such as Red + Bone Meal = Pink Dye.
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There is no 'brown dye' - to dye wool brown, you just place it along with Cocoa Beans into your crafting slot, to make Brown Wool.
You have to put a cactus in a furnace and then apply the resulting dye to the wool.
If you place a red rose in your crafting box, you will receive red dye. You then place the red dye and wool into the crafting box, and you'll receive red wool.
You can craft things with Lapis like: A Lapis Lazuli block, light blue dye, cyan dye, purple dye, magenta dye and blue wool. You can check out Minecraft Wiki for some help and tips.
You can use it on wool, to make a block of green wool, or you can use it on a sheep to make a green sheep, who will yield blocks of green wool when killed or you use shears on it. Cactus Green dye can also be combined with Bonemeal to make Light Green dye.