These are the 23 Biomes in Minecraft: Xbox One Edition
You must travel through at least 17 of them for the achievement to pop. It does count if you are on multiple maps, also you will find every biome in the tutorial level. If you need a picture of each area head over to [ Register or Signin to view external links. ]

1. River (Water only a few blocks deep and a short distance shore to shore)
2. Frozen River (Same as above frozen over)
3. Taiga (Area with only spruce treas and ferns, only spawns sheep)
4. Taiga Mountain (Same as above on a mountain)
5. Cold Taiga (Snowing Taiga)
6. Cold Taiga Mountain (Mountain Snowing Taiga)
7. Extreme Hills (cliffs)
8. Plains (Vast open area with no trees, or very few treas or hills)
9. Ice Plains (frozen over plains)
10. Forest (mixture of many trees, but missing the vines of swamps and jungles)
11. Swampland (short trees with vines and grey colored ground with mushrooms)
12. Swampland Mountain (same as above on a mountain)
13. Mushroom Island (Island with giant red and brown mushrooms)
14. Mushroom Island Shore(Same as above but where the sand meets the water)
15. Beach (Large area of sand that meets an ocean)
16. Cold Beach (Same as above but with snow on it)
17. Jungle (Large bright green trees and bushes covered in vines, coco beans grow on the trees)
18. Jungle Mountain (same as above but on a mountain)
19. Jungle Edge (This is questionable, but its where you will see the forest trees meet the jungle trees)
20. Desert (Large open areas of sand that have cacti)
21. Ocean (Very large and deep patches of water)
22. Nether (Its the nether...how else do you describe it)
23. The End (Where you fight the Ender dragon)

You just have to visit any 17 of these. This can be done in the tutorial world also.

If you need any help feel free to PM me.