For this achievement you need 16 different colors of wool. Some of these wools are easy to get while others are a little tricky. The majority of this will require white wool and bone meal so shear a bunch of sheep and kill skeletons so you have plenty of items when doing this.
1. White Wool
Shear a white sheep
Have 4 pieces of string
2. Red Wool
Get a red flower to create red dye. Combine with white wool to create red wool.
*Have a few extra red dyes for the other wools
3. Yellow Wool
Get a yellow flower to create yellow dye. Combine with white wool to create yellow wool.
*Have a few extra yellow dyes for the other wools
4. Orange Wool
Combine red dye with yellow dye for orange dye. Combine orange dye with white wool for orange wool.
5. Blue Wool
You will need 1 Lapis Lazuli which is underground. combine it with white wool for blue wool.
*Have a few extra Lapis Lazuli for the other wools
6. Black Wool
Shear a black sheep
Get an Ink Sac from a squid. Combine it with white wool for black wool.
7. Brown Wool
Shear a brown sheep
Find a cocoa bean. They are normally found in jungle biomes. Combine it with a white wool to make a brown wool.
8. Green Wool
Destroy a cactus that resides in the desert. Heat it in a furnace for Cactus green. Combine it with white wool for Green Wool.
*Save a couple extra cactus green for other wools
9. Lime Wool
Use some of the cactus green from the last one and combine it with bone meal to create lime dye. Combine the lime dye with white wool to create lime wool.
10. Cyan Wool
Combine Lapis Lazuli with cactus green to make cyan dye. Combine cyan dye with white wool to make cyan wool.
11. Gray Wool
Shear a gray sheep
Combine an Ink Sac with bone meal to make gray dye. Use that and combine it with white wool to create gray wool.
12. Light gray Wool
Shear a light gray sheep
Combine the gray dye you just made on the previous one (you get 2 dyes) with bone meal to get light gray dye. Use that and combine it with white wool to create Light gray wool.
13. Light blue Wool
Combine a lapis Lazuli with bone meal to make light blue dye. Combine it with white wool to create light blue wool.
14. Pink Wool
Combine red dye with bone meal to make pink dye. Combine it with white wool to create pink wool.
15. Purple Wool
Combine a red dye with lapis lazuli to make purple dye. Combine it with white wool to make purple wool.
16. Magenta Wool
*There are a few different ways to make magenta dye. Either option will still get you the dye but will yield different quantity.
Combine 1 purple dye with 1 pink dye = 2 magenta dye
Combine 1 pink dye with 1 red dye with 1 lapis lazuli = 3 magenta dye
Combine 1 bone meal with 1 lapis lazuli with 2 red dye = 4 magenta dye
Once you have your dye, like the others combine it with white wool to make magenta wool.
When you have created all 16 different colors your achievement will unlock. Unfortunately loading a previous save with all of the 16 wools premade didn't unlock for me, even having all of them in my inventory. You have to create them all over again. If there is anything I missed please let me know so I can change it.