Note: I mean the original Crimean war, not the current conflict.
Here's an amateur history junkie (experience: Victoria II, extensive reading of Wikipedia, etc.) giving a short explanation till the experts arrive.
In the words of Shepard Clough, "[the war] was not the result of a calculated plan, nor even of hasty last-minute decisions made under stress. It was the consequence of more than two years of fatal blundering in slow-motion by inept statesmen who had months to reflect upon the actions they took.
"It arose from Napoleon's search for prestige; Nicholas’s quest for control over the Straits; his naïve miscalculation of the probable reactions of the European powers; the failure of those powers to make their positions clear; and the pressure of public opinion in Britain and Constantinople at crucial moments."
Intro:
After Napoleon's defeat and exile, the Great Powers of Europe (Britain, Restoration France, Austria, Prussia, Russia) convened in the halls of Vienna and redrew the borders of the continent, as a child would with crayon a sheet of paper. It was the Concert of Europe, a plan hinging heavily on the international balance of power (no one power is stronger than the other on the continent -- cheeky Britain makes sure it stays on top), committing to paper a system of alignments meant to stop future continental wars (and liberalism, but that's another story) dead in their tracks.
Alas, the Concert of Europe was not upheld for long. Pretty soon most of Europe stopped bothering, and lapsed into an age of pragmatism and war that we see in the 60s, coinciding with German and Italian unification, the Italian War of Independence, the Austro-Prussian War, the Franco-Prussian War, etc.
The immediate cause, at least on paper, was the Catholic French Empire and the Orthodox Russian Empire jockeying for prestige as protector of Christians in the heathen Ottoman Empire. But of course, there was the Eastern Question to deal with.
The rapidly decaying Ottoman Empire threatened to fall at the onslaught of a Russia perceived to be invincible after their victory against Napoleon, and a victory for Russia meant that it would surge to the top and rise in prestige (probably something related to Pan-Slavism as well). France and, most importantly Britain, felt their positions as top dogs of Europe challenged.
After a bout of gunboat diplomacy from both sides, and a race for influence in the Turkish court, Russia intervened on behalf of some minorities, surging through the border and destroying Ottoman ships; this, in turn, brought the intervention of Britain, France, and the rest of the dogpile.
A big cutback to Russian influence and prestige in the region, mainly disarming of the Russian Black Sea fleet, and the perception of the invincible Russian to the paper tiger. An analysis of military bumbling on both sides, most notably Britain and the Charge of the Light Brigade. France being recognised as protector of Christians in the Ottoman Empire. Russia probably lost some land.
Above all, the admittance of the Ottoman Empire, a foreign foe, the Oriental Ishmael with heathen beliefs, into the Concert of Europe, while Russia was relegated, for a period, as a Secondary Power. Yada yada, Ottoman territorial integrity, increasing Balkan autonomy.
In Russia, paved the way for sweeping reforms and liberalism under Alexander II the Liberator (abolished serfdom, pretty cool guy). A weakened and politically isolated Austria and Russia allowed Italian Unification (dudes were allies of France) and a resolution to the German Question (Kleindeutschland (Lesser Germany) under the hegemony of the Konigreich Preussen (Prussia)), therefore the Franco-Prussian War, German Unification, enmity over Alsace-Lorraine, etc.
Then there was nationalism on the rise in the Ottoman Empire (which culminated in Balkanisation throughout the next decades), Austria (you can tell it didn't go well when they renamed themselves to Austria-Hungary, and later on when some sherbet shot a hungry ostrich called Archieduke), which led to further Balkan crises involving Russia, and something about Pan-Slavism, Bulgaria, and Serbia.
Then there was a rematch in '77, ending with a close Russian victory.
With the Concert of Europe and its balance of power system broken, the world saw France leap its way up, only to be supplanted by the rising North German Confederation over a succession crisis in Spain manipulated by Otto von Bismarck. Britain leered across the Channel in fear of the hegemon that dared take its rightful place as overlord of the continent as Napoleon, and many others, had dared to attempt in the past.
100 years after the convening of the Congress of Vienna, the lights went out all over Europe. The guns of Sarajevo blazed o'er the din, and before long the world was plunged into THE war.
While I'm no historian, I can say that one of the outcomes of the Crimean War was the sale of Alaska to the United States. Russia incurred so much debt during the war, and they had this big piece of "worthless" land that would have cost a lot to defend. To rid themselves of the problem, Tsar Alexander II saw an opportunity to make some cash, and sold Alaska.
One of the things that I think is also interesting about the Crimean war was it's use of the telegraph. No doubt that the telegraph was used for tactical purposes (though I have no sources to back this up), but, perhaps more importantly, it was used for journalism. I don't think it is too much of a stretch to say that the Crimean War was the first "live" reporting war. Journalists associated with the United Kingdom reported updates on the war when they could. Ultimately, this lead to the people learning how horribly mismanaged the war was, see: The Charge of the Light Brigrade where British light cavalry charged head-on into dug in Russian soldiers, and saw over 250 casualties on the British side, with very little losses on the Russian side. The ill-fated charge would later be immortalized by Lord Tennyson in a poem and as a song by Iron Maiden.
Hopefully someone else can address the roots and causes of the war, because the only thing that I know regarding the Crimean War was that the Russians were expanding, and, at the time, there was this thing called The Easter Question which was how the western powers (France, UK, Prussia, etc) viewed the ongoing tensions between the Russian Empire and the Ottoman Empire as well as the internal issues that the Ottoman Empire was having in the Balkan Peninsula.