Single Player "World Completion" Achievements:
There are 5 worlds with 6 levels per world. The names of the worlds, in order, are Skyworld, Iceworld, Caveworld, Cloudworld and Lavaworld. Note there is no acheivement for beating Cloudworld (World 4.) This is the only world without a specific achievement. Note that the Skyworld achievement is unlike the other "World" achievements in that it only asks for you to complete the first 3 levels. There is also one miscellaneous achievement in the Single Player mode called "Spyglass." That is covered at the end of the guide and not in the Single Player section.
For this achievement you have to beat the first 3 levels of the game (1-1,1-2,1-3.) These 3 levels are introductory levels and shouldn't cause to many problems. There is no time stipulation for beating the levels, simply finish them as fast as you can. They are only the first half of Skyworld and there are 3 more levels in this world. You can go to the next world (Iceworld) after beating Level 1-3 but you should probably just beat all 6 levels, trying to do them all in Silver Medal time (see "Medal Achievements below) and then continue on to Iceworld.