There have been questions concerning battles, dislodgement, and convoys. I'll attempt to go over the basics of each here.
Battles:
The battles of diplomacy are quite different than the battles of other strategy board games like Risk, A&A, etc. The latter games involve 2 units squaring off against each other, with one the victor and the other removed, while Diplomacy is more like jockeying for position. As I mentioned in the "basics on moving and supporting" it's easier to think of attacking as pushing or simply moving. If you want to move into a space, you order the unit to move there. If know you will encounter resistance (or think you might) you can use your own units (that are in adjacent spaces) to support the move, or enlist the support of friendly units belonging to another player.
Some things you need to remember: only one unit in a space at a time. All units have the exact same strength. So it's basically strength in numbers. If you have more units supporting you than your opponent has supporting him, you get what you want (i.e. you move into the space if you were trying to move, or you stay if you were trying to stay.) You can have units support you if you just want to stay in a province too, and think someone might try and move into it.
Dislodgement:
When a unit gets pushed out of a space, it is dislodged. It is not removed from the board or "killed," it is just pushed out. The owner of the unit that is pushed out has to decide where the unit gets pushed to, and there are a couple of rules concerning this. One, for example, is that a unit that gets dislodged cannot retreat (which is what happend when you're dislodged: you retreat) into the same province that the attacking army came from, the "attacking army" being the army that pushed you out. You cannot retreat into a province that is already occupied, either (only one unit per space.) If there's just nowhere to go, then the unit is removed from the board. A player can also choose to voluntarily remove a unit instead of retreating, even if there are places to retreat to.
Pushing units out of space is important, because the goal is to occupy the supply centers. The number of supply centers you occupy at the end of a Fall turn determines how many units you have. That's how you "kill" units: you occupy your opponent's supply centers and cause him to have to remove units after a Fall turn.
Convoying:
Convoying is using ships to move armies across the water. As you are aware, an army cannot occupy a sea zone. So to move an army from London to Belgium, for example, you need a ship in the sea zone between the two land spaces (either the North Sea or the English Channel.) You write the order for the army to move to the desired space (A Lon-Bel, or Army in London moves to Belgium.) Then you write the order for the fleet to convoy the army (F Nth C A Lon-Bel, or Fleet in North Sea convoys the army from London to Belgium.) The army will then move to Belgium.
That is, unless the move is contested. Regular rules about pushing apply, and if someone pushes against your convoyed army, and they push harder (have more support) than you, your army will not be able to convoy. You can have units supporting your convoy, so you can push harder also. Just remember that as in regular conflicts, whoever pushes hardest in convoys wins. Also remember that equal forces repel. A fleet convoying an army does not count as supporting that army, so if an army is being convoyed into a space it is pushing with a strength of 1 (itself.) If that space is already occupied, and the occupying army isn't moving anywhere else on this turn (it's holding) then the convoy will fail. (1 vs. 1 = defender stays.)
Convoys can also get broken up if the fleet that is convoying the army gets dislodged.
Notice that convoying an army takes up the whole action for a fleet on a turn. Each unit can only be ordered to do one thing per turn, and so you cannot ordera fleet to convoy, and then move somewhere else on the same turn.
