In Diplicity players submit orders by using the map item in the game navigation menu. To understand the exact meaning of the various orders, see https://en.wikibooks.org/wiki/Diplomacy/Rules or, for the technically interested, http://web.inter.nl.net/users/L.B.Kruijswijk/.
Most variants add their own set of rules to the standard Diplomacy rules, and these can be seen by clicking on "Variant information" in the game menu. If you have questions which aren't answered there then please ask in the forum.
In the map view, any province expecting an order from the player will be highlighted. Clicking a highlighted province will bring up an order selection dialog, and further selections will continue to ask for order input in the form of provinces, or unit types.
A player that is satisfied with their currently submitted orders can tell the server that they are ready for the game to enter the next turn. In the Android client, go to the phase settings item in the game navigation menu and check RDY.
When all players are ready, or the phase deadline runs out, the phase will be resolved.
A player having being ready before phase deadline will get an increased Quickness and Reliability rating.
A player having submitted at least one order, but not declared that they are ready, will get an increased Reliability rating only.
Hint: In almost all views inside a game in the Android client, swiping quickly left or right will move one phase back or forward in the game.
Note: the map for the Classic variant is an adapted variant based with permission on a version by /u/trampolinebears.