The objective of the game is to be in possession of 18 supply centers. The player who does that wins. Generally though, a draw is agreed upon before that happens. Supply centers are the spaces that are colored. Neutral supply centers ones are light blue, while others are owned by players (yours is green). White and sea spaces are not supply centers.
To place ownership over these supply centers, you must have a unit in the space until winter rolls around. Units come in two types: armies and fleets. Armies move over land, fleets move over sea and coastal land spaces (land spaces that are adjacent to the sea). Fleets can also convoy armies. They can send an army across the sea to another land space, provided that the army and its destination are adjacent to the fleet's space. Convoys can also be chained if there are other fleets nearby. All units can only move one space per turn, with the exception of convoyed armies.
Like a lot of board games, Diplomacy is split up into turns. There are three major turns each year: spring, fall, and winter. Spring and fall are movement turns. However, winter is different. Winter is when you build or disband units according to how many supply centers you own by then. i.e. If you lost a supply center, you lose a unit. If you gained two supply centers, you build two units. You can only build in your home supply centers, which are the supply centers you start out in. After winter, a new year starts over again.
As for the movement turns, there four types of orders you can make during them: hold, move/attack, support, and convoy. Hold is the default order if the player doesn't send any for the unit. What it does is simple: nothing. It stays put. Moving is also simple: the unit moves to an adjacent space. If there isn't another unit there, then the move is a success. Buuut if another unit moves there as well, neither of them get in (called a bounce).
This is where support comes in, which is where the game can get a little complicated. Units can support other units into a space (the supporting units must be adjacent to the space that the other unit is moving to), thus forcing out any opposing units in it and making them retreat. This is called offensive support. However, if the opposing unit has defensive support from equal or more units than the attacking unit, then nothing will happen and everyone remains where they are. Units that provide defensive support must be adjacent to the unit that they're giving support to. The amount of support decides the result. i.e. Attacking unit is supported by two other units. The defending unit is supported by one other unit. The attacking unit takes the defending unit's space because 2 > 1. If the defending unit had support from one more unit, then the attack wouldn't have been a success (2 = 2).
I know that's a lot to digest, but hopefully I explained everything well enough without being too lengthy. That's pretty much all of the rules, but you can read more about the finer details
here, like examples of how to organize and abbreviate your orders.
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