These achievements are awarded for defeating each boss in the game, one achievement for each boss. The final achievement, the "World Conqueror," will unlock after you defeat the game's final boss, the Dark King. Bosses are the driving force behind the game and act as the only hint of storyline you will get. Defeating them should motivate all your actions in the rest of the game. Essentially you just want to level up so you can get a more powerful army to take down the bosses.
Boss fights are fairly lame in this game. You never actually fight a boss, but just a series of normal soldiers which I guess could be considered minions. Each boss battle has three areas with three rounds each (except General McCallister which has 4 areas and the Dark King which has 5 areas). This totals nine battles to defeat most bosses. The enemies will get more challenging as you progress. At some points, you may not be able to defeat the enemies you are facing using normal tactics. Take this time to grind a few levels to get better units. If you cannot start the battle and it says you must unlock a specific quest, this generally means you need to talk to Shaara or Lisa (the fictional neighbors in the game). Do this by visiting their respective kingdoms or, in some cases, by just clicking on the quest in your task list.
FINAL BOSS: The final boss consists of 5 rounds of 3 battles against some relatively challenging opponents. These enemies will occasionally be those that you can not even recruit yourself. They can only be purchased with diamonds. The enemies will also almost always be fully-upgraded as well. You will probably need to upgrade the highest ranking units you can purchase. It is a good idea to generously use whatever spells you may have accumulated throughout your playthrough. Use the diamonds you have accumulated wisely to purchases additional spells if you need them. If you saved all diamonds that you could, you should have around 80 to work with. Some battles are quite one-sided and it might be hard to compete without using spells throughout the battle.
See the "General Tips" section in the Roadmap for more tips on succeeding in the game in general.
*GLITCHY WARNING* Each and every boss achievement in the game can be considered glitchy because sometimes the achievements will just not unlock after completing the requirements. To prevent having to worry about this, right before fighting the last battle against any boss (i.e. before the 9th battle), pause the game and click "Save" to save your game to the cloud. After defeating the boss, if the achievement doesnt unlock AND stick to your xbox.com profile, immediately exit the game and then load it back up. The game will prompt you to choose your local save or cloud save. Choose the cloud save and then battle the boss again in hopes that it will not glitch once more. If it does, repeat the process. This fix requires an active internet connection.