War is a game mechanic where corporations and/or alliances can fight in hisec or lowsec space without CONCORD intervention or security status loss.
War Declaration Mechanics ('Deccing')
To initiate war, a corporation/alliance must pay a fee to CONCORD. To declare war on another corporation/alliance you must have the director or CEO role in your current corporation.
The CEO or a director of the Corp (Executor Corp for alliance) must open the Corporation interface, go to the Alliances tab (in case of alliance), then view the "Home" tab. At the bottom of the window is button labelled "Declare War". The CEO must push the button and enter the name of the target corporation or alliance, then press search. Once he or she has confirmed the selected alliance is the correct one, war is immediately declared, with fighting becoming possible after a 24 hours notice period.
Once war is declared, there is a 24-hour startup period where fighting cannot occur, to give both sides a chance to get ready for the war. Once the war is active, there will be a period of one week where any aggression can occur between the two sides without CONCORD intervention.
Costs of wardecs
Wardec costs at least 50 million isk for a corporation to declare war on another corporation. The war declaration fee must be in the Master Wallet of the corp wallet.
The formula for calculating wardec cost is (log2.05831 N)^2 * 300000 * N^0.27, where N is the number of corp members. The cost is now the same when declaring on a corporation or an alliance. 
This base-cost can be increased by the number of wars you currently have ongoing (where you are the aggressor) and the number of wars the target corporation already has against them. The formula for calculating the cost is B * (N +1) * (W + 1). Where B = Base Price, N = number of wars you currently have, W = number of wars currently against the target corporation. Mutual wars are not considered in the cost calculation.
If the receiver of the wardec chooses, they can make the war dec mutual. Mutual wars do not cost the aggressor the usual war weekly fee and won't expire unless the war is manually retracted or the mutual status is revoked. Mutual wars also do not increase the cost of declaring war on other organizations or the cost of an organization declaring war upon your organization.
Ending the War
Wars can end in one of several ways:
- If the week ends without a renewal fee paid for the war, then CONCORD will declare the war invalid.
- War can be surrendered by the offending/defending corporation's CEO or Director. Other side may accept or decline it.
The surrender option is streamlined (no more meeting in same station) and can involve formal concession of ISK. It also means these entities cannot war dec each other for the next 2 weeks.
Whichever way a war ends, there will be a 24 hour cooldown period where fighting can still occur. This prevents ending a war to avoid a particular engagement. After this cooldown, normal CONCORD services will resume.
University Guidelines during War
EVE University used to maintain a different set of rules during war, which in various iterations were very restrictive to fairly relaxed. Under older mechanics and an E-UNI scope far less than it is today, these were designed to make conflicts as un-fun as possible for the aggressors, and to reduce the number of wardecs that the University received. Probably the most important pages to read are the the Wartime SOP, Wartime Operations in EVE University, and the Tips For War. Listening to a War 101 class and browsing the rest of the wiki is highly recommended.
How to Check the Status of a War
Open the corporation window in your NeoCom and select the Wars tab. Note that fighting can occur 24 hours after the war started and 24 to 25 hours after it is finished (the 'can fight' column shows, if the war is live).
You can also use the "Show Info" on any given corp/alliance and go to the War tab and you can see the history of finished wars, current wars, and pending wars.