Hopefully a relevant flair like /u/facepoundr will contribute their take.

Until then, I'd suggest you check out Orlando Figes' work especially The Crimean War: A History. It's a massive book, but it really goes into the appropriate context.

It's difficult to sum up. Essentially, the venerable Ottoman Empire was dying. It was widely seen in Europe as hopelessly outmoded and its dominance over Christians as unacceptable in the long term. All the Great Powers were determined to carve out a piece of the empire for themselves and to stop the others from getting what they wanted. In the 1850s, Russia was ascendant. They were the key to crushing the liberal revolts of 1848 and their empire spread steadily into Central Asia. Intellectuals in Britain and France increasingly saw them as a threat.

The British felt threatened by the Russian expansion toward India and intrigue in Persia. The French, the major holders of Ottoman debt, felt threatened by Russian intentions there. Further the French declared themselves the protectors of the Catholic population of the empire, much the Russian tsar had declared himself the protector of the Orthodox. The two churches contested the Church of the Holy Sepulchre in Jerusalem between themselves. The Sultan was French educated and favored their control of the site. The Tsar saw this as an insult and moved troops into the buffer states where today exists Romania. The Ottomans eventually responded with a declaration of war.

The French and British wanted to defend the Ottomans, though more to contain the Russians and preserve their influence in the empire. Despite Allied worries, the Ottomans crossed the Danube in 1853 and had some success against the Russians. The Russians beat them back and crossed in turn in the spring of 1854. They hoped for a rebellion of the Orthodox Christians there to support them, but were disappointed. The Austrians unexpectedly entered the war on the side of the Turks and invaded Danube border states. This forced the Russians to retreat in June, 1854, to avoid their supply lines being cut.

French and British forces finally arrived in theater that very month, only to see the Russians pull back. The Allies decided to crush the center of Russian power in the region, the naval base at Sevastopol. They landed north of the city that September and defeated the Russian army at the Battle of Alma. They circled around to the south to avoid the worst of the Russian defense and began bombarding the city. The Russians received reinforcements, however, and decided to try and break the siege.

This is when we get the famous Battle of Balaclava and the less famous Battle of Inkerman. Essentially, the Russians were thrown back into the city. Then followed the long and bitterly cold siege of Sevastopol. Bombardments, disease, and starvation killed many on both sides. Russians attempted to break out a few more times with varying success. Finally, they decided to destroy what they could retreat to the north in September 1855.

The Russians had more success in the Caucasus, eventually capturing Kars and a large part of eastern empire. Things went less well for them in the Baltic where were blockaded and shelled and their trade ruined.

By 1856, everyone wanted the war finished. All territory was restored to its pre-war possession, but the Russians were barred from militarizing the Black Sea again. The Danube states, Wallachia and Moldavia, were given more autonomy. Essentially, Russian ambition had been stymied.