A war coalition is a set of countries fighting a particular war together. It is a temporary form of alliance which lasts only while the war lasts. War coalitions are created from the members of a military alliance, but as the war evolves, the coalition may add and remove members independent of the parent alliance.
Members of a War Coalition
When a war is declared, only the country declaring war and the one declared against are initially in their respective war coalitions. When a country makes an alliance call, all members which honor the call join its war coalition. The offensive alliance gets called first.
A war coalition can add members after the initial DoW. This happens any time a country joins the military alliance of the war coalition's war leader. (A country joining a second country's alliance joins all ongoing wars which the second country is leading.)
War coalitions lose members any time one negotiates a separate peace with the enemy coalition or with a member of it. A member is also lost if it ceases to exist.
Each war coalition has one member which is its war leader. Only the war leader is allowed to negotiate peace for the coalition; other countries in the coalition can only negotiate separate peaces.
When a war is created, the alliance leaders of the two sides are initially the war leaders of their side. (If the existing alliance leader dishonors the alliance call, some other country will become the alliance leader; and that country will lead the war coalition if it joins the war.)
Once a war exists, the leadership of a war coalition can change only if the current war leader makes a separate peace in that war, or if the war leader ceases to exist (i.e. was annexed by a third party). In this case, war leadership will devolve to another member of the coalition.
Seeing War Coalitions
The set of war coalitions currently in existence can be viewed on page 20 of the ledger, entitled "Current Wars". This page lists all current wars, regardless of whether your country knows of the countries involved. The format of the listing is as follows: first comes the boilerplate text "War between". Then the aggressing war coalition is shown, the text "against", and finally the defending war coalition. The war leader of each coalition is always listed first.
You can also see the wars a particular country is involved in via the diplomacy screen. This listing does list war leaders first, if they are not the country you are looking at. But you can not see the offensive/defensive nature of the war.