A new addition to the game with the Heart of Darkness expansion, the crisis system is a way for smaller, weaker nations to free their people from the oppression of greater powers. By starting a crisis you threaten to bring the whole world crashing down in a war over your land, so be wary.
Starting a Crisis
A crisis can begin randomly in any flashpoint province with enough tension. A flashpoint province is any that has the cores of another nation, or pops of that nation's culture. Tension is generated by liberation or unification movements, and can also be increased by a national focus placed in another nation's territory for regions that have your cores. This focus is only usable by nations who are not Great Powers. So it's not possible to take Mexico's Manifest Destiny provinces in this way.
Tension is also generated by two nations colonizing the same provinces; as the level of colonial points invested by both sides increases and neither draws ahead then the area becomes a flashpoint and tension is generated, until one side backs down or a crisis is caused.
Resolving a Crisis
When a crisis begins all great powers can attend, unless they are at war. Great powers on the same continent will suffer a prestige penalty for not attending, but others on different continents will have no obligation. Each side in the crisis must seek a great power to back it, unless one of them is a great power, in which case they will back themselves.
From there the other powers can pick a side. Picking a side is the best option if you do attend, otherwise there is a prestige penalty for not doing so. Nations played by the AI will be reluctant to involve themselves early in the crisis, so the first nations to take sides will likely be the stalwart allies of the side they take.
If a neutral nation does not lean heavily one way or the other, the leading nations may wish to offer them various concessions to induce them to participate. These concessions are available for the leaders of the two sides to offer starting from the beginning of the crisis and, if accepted, will be added to the goals of the side they join; if a nation does not get what they asked for out of a crisis, then relations with them met will suffer.
Right from the start the powers backing either side can make offers to solve the crisis, though a rejected offer will make the crisis "hotter" and therefore closer to war.
If this wheeling and dealing should fail then there is only one recourse: war. A war will be fought until one side or the other gives in and a peace is signed. This can bring all of the great powers in the world into a conflict without an alliance, so it is best to be vigilant and ensure good relations with anyone you want to particularly avoid fighting.
After a side succeeds in a crisis, the leader's relations with all nations which gained something they wanted, or joined without asking for anything, will increase. A leader's relations with nations which do not get what they want, or with all of their allied nations if they capitulated, will suffer.
In the event that a crisis is not resolved diplomatically, the opposing sides fall into a crisis war. These wars function similarly to other wars, however in crisis wars no separate peace treaties can be signed, meaning that all belligerents in both factions must remain in the war until one side surrenders.
Common Crisis Locations
The Balkans - All of these nationalities, Serbs, Bulgarians, Greeks, Bosniaks, and Albanians, languishing under the rule of the Ottoman Empire can cause crises with alarming regularity. Particularly in the late game when militancy has built up. Liberation movements will spring up demanding freedom, and the Turks will be hard pressed to hold them back.
Poland - A relatively populous area, Poles have cores in Prussia (which will likely become Germany later), Austria, and Russia. This means that any large liberation movement can cause a crisis for any of these three countries. Fortunately for someone playing as any of these nations, you will probably be backed by the other two as they have no wish to see a resurgent Poland with cores and a culture CB on them.
Korea - In Heart of Darkness Korea begins as a free nation, opening them up to quick Japanese conquest. Beware however: the millions living there will resent your rule and with so many people the tensions rise awfully fast.
Washington/Oregon - These two uncolonised provinces are both desired by the British and Americans. While the British will normally simply ignore Idaho, they will often send an expedition to Washington, and, if that is uncontested, to Oregon This can cause a crisis early on, so USA players are advised to spend their diplomatic points seeking a backer on the continent, as a war with the British so early will go badly. Alternatively just withdraw, and claim it later as part of your Manifest Destiny.