The Papal States had their hands full at home. Pius IX was the pontiff from 1846 to 1878, so there was only one papal head of state for the American Civil War. The (often violent) process of Italian unification was underway during Pius IX's pontificate. Many date this process back to 1815, but it was surely underway in 1848 when the pontiff had to briefly flee Rome due to a revolution. The Second Italian War of Independence began in 1859 (featuring France and Piedmont against the Austrian Empire) brought conflict in Italy to the pontiff's doorstop. By 1861 the Papal States had lost all of their territory except Rome itself--and that was guaranteed only by the presence of Napoleon III's French troops. If we cut off our story in 1865, that situation for the papacy would remain unchanged.

You may read from time to time that the Vatican (or the papacy, depending on the source) recognized the Confederacy. Pius IX did write a letter to Jefferson Davis. In that letter, he did address Davis as the “[I]llustrious and honorable President of the Confederate States of America.” However, the full reality is not nearly so dramatic. This is the only place online that I have found the letter from Pius IX to Davis. It is a PDF, and it also comes from a less-than-stellar source. Ignore the rest of the content, just scroll down to the text if you want to read the letter. In sum, the letter is the equivalent of a form letter. It acknowledges receipt of Davis’ earlier letter, says that it would be really nice if there was peace instead of war, asserts that the pope will pray for peace, and that he will also pray that Davis will be granted grace.

That’s it.

So, why the address as “President?” Basically, the pontiff filled in the blank by looking at the signature of the original letter from Davis. With Italian unification and being a self-proclaimed “prisoner of the Vatican,” Pius IX had much greater concerns much closer to home than the Confederacy. Also, the idea that the pope would somehow back the South in the US Civil War is far-fetched, given the less than ideal way Catholics were treated in that region. So, if you hear someone claim that the pope recognized the Confederacy they are incorrect. The truth was much more banal—the pope just replied to Davis’ letter with bland pleasantries, largely because events closer to home dominated his attention.

The French under Napoleon III was inclined to intervene, if nothing else then by breaking the USA blockade of the CSA. However, Napoleon III would not undertake any kind of action at sea without the approval of the British. The British were less than lukewarm about the idea, and thus no intervention came about from the French. The French did extend credits to the CSA to some extent and did build the ironclad CSS Stonewall for the CSA, but otherwise did not interfer. They were, as /u/Domini_canes describes from the other side, involved in the unification of Italy at the time. The French also got involved in Mexico by placing Archduke Maximilian (brother of Emperor Franz Joseph of Austria) as Emperor in Mexico. In that they had support from Austrian and Belgian (Maximilians wife Carlota was Belgian) volunteers. This intervention was in breach of the Monroe doctrine, and would most likely not have been possible if the USA had not been in a civil war.

The British toyed with the idea of intervention, but the general populace of Britain was staunchly abolotionist. Britain had spearheaded the abolition of the slave trade in 1807, declaring slave shipping the same as piracy and establishing the West African Squadron to combat slave trading 1808. Slavery was abolished in the British Empire 1833 (except for India, Ceylon and Saint Helena, where it was abolished 1843). Popular sentiment made it pretty much impossible for the British to side with the slaveowners of the CSA, and barring a major incident, intervention was far away. The Trent Affari of 1861, when a USA ship boarded a British mail ship, the RMS Trent and took her prize because she had two CSA diplomats bound for Britain aboard did cause an outrage in Britain, but Lincoln ordered all British demands agreed to, and Prince-consort Albert (husband of Queen Victoria) did much to soothe the sentiment in Britain, and the affair blew over. The British did sell arms and ammunition (primarily Enfield rifled muskets and Armstrong cannons) and bought other goods through the USA blockade, did extend credits to the CSA and did build and equip ships for the CSA navy - the latter which they paid reparations for after the war.

The Austrians were still licking their wounds after their defeat in the War of Italian Unification against Sardinia-Piedimonte and France, and were in the process of reforming their army. They were incolved in the French attempt to place Maximilian on an Imperial throne of Mexico but also sold large amounts of Lorenz rifle muskets to both sides in the war. Austria would send a large part of its small navy to fight against the Danes in the North Sea 1864, as the Prussian navy was almost non-existant.

Prussia were at the time gearing up for the war against Denmark 1864 and eventually the struggle with Austria over the domination of Germany and with France over the unification of Germany., Generally, the Prussians, who had learned and improved the French staff system were disdainful of the amateur style of American warfare. There's an oft-quoted statement by von Moltke the elder that the US Civil War "consists of two armed mobs chasing each other around the country, from which nothing can be learned", however, I have never been able to substantiate it in any sources, so it is probably made up. However, it does give the sentiment of many European professional officers - their own wars were rapid and decisive with heavy pursuit by reserves and above all cavalry of the defeated army, rendering it unable to fight again - the inability of the USA or CSA armies to do that made European officers view the war as little to learn from.

Russia supported the USA and maintained friendly relations - a large part of the Russian navy visited USA ports, partially as the Rising in Poland had the French talk about another Crimean War and the Russians wanted to maintain a part of their navy outside the Baltic and the Black Sea.

Sweden firmly supported the USA and large amounts of Swedish emigrants Went ot the USA and many of them volunteered to fight for the Union side.

The war was really important to Europe, particularly Britain and France, which relied on the trans-Atlantic trade and particularly the textile industry. By the summer of 1862 one-half of British textile workers were unemployed. Government revenues from textile production, and in total, declined. The French foreign minister said "We are nearly out of cotton, and cotton we must have." (Jones, Union in Peril)

There was real pressure to subvert the blockade, and thus deal with the Confederacy directly and risk war with the Union, but this was obviously a last-resort option, especially when the status of the Confederacy was still... doubtful.

It was a cool decision point, where the European powers could either wait out the war, and bear witness to a "bloody and purposeless war," or subvert the blockade and risk war with the Union. William H Seward did a good job of dissuading the later course of action, and the new cotton production in Egypt and India offset the loss of supply from the South.

There was also the whole issue of slavery but I'm not really up on that and I'm sure someone else is.

Most of the above from George C Herring's From Colony to Superpower.