The Sims 3

This is probably the easiest achievement in the game. Press Back and go to the build and buy menu. When you enter it push Y and go down to create a style. From there pick any piece of furniture, wall or flooring that you want. There will be a couple options to change the color but you're going to want to scroll over to the flower looking symbol option and enter that. From there you can edit the color, material and pattern of the thing you've chosen. Once you've chosen a new color, material or pattern press Y to save the style. Once the style is save move on to a different color, material or pattern. This can be done on the same piece of furniture so you don't need to worry about finding 20 different pieces of furniture. Once you've saved your 20th style you'll unlock the achievement.

While other solutions have suggested using furniture, floor, or walls there is even a quicker way. When you're given the chance to style your Sims 'work' clothes just go into it and pick either top, shoes or pants/dress and hit Y and then pick out 20 different colours/designs (has to all be different) and remember to hit Y to save each different one. Just doing this x20 will make the achievement pop-up.

I did this with my Sims work shirt. I just changed the main part of the shirt 20 times then achievement popped up. Very quick and simple.

Go to build mode and press "Y", click the "Create a Style" option and choose a bed or a chair.
Change a colour and save the game, do this 20 times for the achievement.

Have not tried saving the same pattern 20 times.
Clothing styles do count apparently so you could get this real early when you're making a character.