Source: 1789-1900 European history course from a few years back

The Ottoman Empire was in a state of decline for quite a while by this point; it was called the "sick man of Europe". Basically the three greatest European Powers at that time - France, Russia, and England - all had a stake in the region.

For the Russians, the demise of the Ottomans would be a good thing, because they really, really wanted access to a warm-water port (one that didn't freeze in the winter) and they also wanted to have access to the Mediterranean for both economic and military reasons.

Meanwhile for the English and the French, the continued survival of the Ottoman Empire was the goal, because it acted as a bulwark holding back Russian expansion into the region.

Furthermore, following the defeat of Napoleon in 1815, the Great Powers were really reluctant to let one country upset the balance of power, in fears of repeating the rise of another Napoleon and the start of another big war.

Other reasons included France's (or perhaps more accurately, Napoleon the 3rd's) ambitions to relive the glory days of the Napoleonic Empire, and England's direct interests in maintaining naval control of the Mediterranean.

As for the actual circumstances under which the war happened, I don't know many details so maybe someone else can fill that in. All I know is it was religious - the Ottoman Empire included the Holy Lands (Jerusalem, etc) and both parts of the Christian Church - Roman Catholics (backed by the West) and Orthodox (backed by Russia) - wanted control over these places. They've been fighting for religious control over a long period of time, and by the 1840s the Orthodox Church was in control of the Holy places, while the Roman Catholics were trying to get it back. The French/English convinced the Turkish Sultan to deny the Russians' demands to have control of the holy places, protection of Orthodox Christian subjects within the Ottoman Empire, as well as some other concessions.

In response, the Russians invaded Moldavia/Wallachia (autonomous parts of Ottoman Empire) to test the Sultan's resolve - they didn't expect England and France to respond but they did - and the war started.