In the biblical story of david and goliath david the hero of the israelites defeat a fearsome giant based on what you know about the crimean war who does the giant in the cartoon represent

 RUSSIA

I've attached an image of an 1854 cartoon published in the French magazine,  Le Charivari.   I suspect it's the cartoon your question has in mind.  The Russian Empire, personified in the cartoon by an image of Tsar Nicholas I, is pictured as Goliath.  A generic Turkish figure (as David) represents the much smaller Turkish Empire.  

The European powers in general had feared a war with Russia, thinking the gigantic Russian Empire could ultimately outdo any other power.  But the Crimean War showed that the Russian Empire was not as formidable as had been thought.  The Turks, along with allied forces from France, Britain, and Sardinia, were able to defeat the Russians.

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