The charted warp device helps you move your city on a new location which you select the coordinates for. To use a Charted Warp Device, you need to select a swamp and then click "warp."
The Dark Warp Device can also change the location of your city; however, this is a random location warp. To use a Dark Warp Device, just click it in the "General" section of your inventory.
If you are being bullied or repetitively attacked by certain players/alliances, you may want to move. These players can always save your coordinates in their messages, thus will know exactly where you are. By using either device, this knowledge will become obsolete; although, this does not promise that these players will not scan the map for your city again.
The Outpost Warp is used to move any of your Outposts. In order to warp an outpost you need to own a plain. It is possible (but not recommended) to warp to a lower level plain.
Keep in mind that you don't want your Outposts too close to your city! In war, players may aim for more than just your city, so you want to make them harder to find. The most important Outposts to "hide" are the Gaea Springs and the Cliffs of Chronos. The Amber Crest Dragon enables to you obtain an incredible 10,000,000 resources in just one march, and they have an especially long healing time- nevermind if they're actually attacked! The Chrono Dragon also has a lengthy healing period, but is useful in terms that any march led by it will take no more than a minute, regardless of distance, so is convenient for quick attacks. You don't want to have these Outposts set alight!
You cannot warp your city if you have any troops marching, even if they are returning to your city.