Many foreigners fought in the American Civil War. Often in their own regiments both in the North and the South.

In the North one reason for this was the way the Enrollment Act was written. One could serve, pay $300, or hire a surrogate to serve for you. This wording inspired lucrative businesses in recruiting surrogates. Recruiters would meet ships on the docks to enroll immigrants. The argument was if you volunteered to be a surrogate you were paid if you didn’t and were drafted you would lose out. The immigrants who often didn’t have the money to opt out often accepted money to opt in.

Another reason for the high numbers of foreigners in the American Civil war was the desire for Military training. Many of these troops like the Irish, Italian, and German were sympathetic to revolutionary movements in there own homelands. It’s a central theme of the commanders listed be low. A thought was they would take there experience in the American Civil War and return home to fight there own revolutions. This did not occur however likely because the American Civil War was so costly in blood to those who fought it most were either consumed or had there fill of fighting when the war was concluded.

The Irish

The first to come to mind were America's largest ethnic block at the time, The Irish fought on both sides of the American Civil War.

More than 150,000 Irishmen, most of whom were recent immigrants and many of whom were not yet U.S. citizens, joined the Union Army during the Civil War.
History.com

The Irish fought integrated into American regiments, but they also formed all Irish volunteer regiments which together were called the Irish Brigade.

The Irish Brigade included:

• 63rd New York Infantry Regiment
• 69th New York Infantry Regiment
• 88th New York Infantry Regiment
• 116th Pennsylvania Infantry Regiment from Philadelphia,
• the 28th Massachusetts Regiment

The Irish Brigade often fought at the vanguard, and the resulting higher casualties some attributed to discrimination were among the motivations of the NY Draft Riots of 1863.

With Regards to the Irish Brigades high losses, and being central to many battles.

This meant that they suffered disproportionate numbers of casualties. At the Battle of Antietam, in September 1862, about 60 percent of the soldiers in the 63rd and 69th New York regiments, almost 600 men in all, were killed in battle. A few months later, at the Battle of Fredericksburg, 545 of the brigade’s 1,200 men were killed or wounded. “Irish blood and Irish bones cover that terrible field today,” wrote one soldier. “We are slaughtered like sheep.”

In July 1863, at the Battle of Gettysburg, about 320 of the Irish Brigade’s remaining 530 soldiers were killed. (There is a monument to the Irish Brigade on the battlefield there: a green malachite Celtic cross with a trefoil, an Irish harp and the numbers of the three New York Irish regiments rendered in bronze on its front. At the cross’s feet lies a statue of an Irish wolfhound, a symbol of steadfastness and honor.).
History.com.

The Irish Brigade was lead by Brigadier General Thomas Francis Meagher.

In February 1862, an Army captain named Thomas Francis Meagher became the Brigadier General of the nascent Irish Brigade. Meagher was born in Ireland, where he had been active in the “Young Ireland” nationalist movement and exiled as a result to the British Penal Colony in Tasmania, Australia. He escaped from Australia in 1853 and came to the United States, where he became a well-known orator and activist on behalf of the Irish nationalist cause. He joined the Army early in 1861. Meagher was ambitious, and he knew that if he could raise an all-Irish infantry brigade, Union Army officials would have to make him its commander. He also hoped that an Irish Brigade in the U.S. would draw attention to the nationalist cause at home.

After the Civil War, Thomas Francis Meagher became the Acting Governor of the Montana Territory. He drowned in the Missouri River in 1867.
History.com

The South also had an Irish Brigade lead by General Michael Corcoran

Few know that Irish immigrants played an equally important role in the Southern Confederacy. Over 40,000 Irish fought for the Southern cause. They were the largest immigrant group in the army, and they made up about 10% of all Confederate combatants. Sligo Heritage

The Germans

Major General Franz Sigel

Franz Sigel (November 18, 1824 – August 21, 1902) was a German military officer, revolutionist and immigrant to the United States who was a teacher, newspaperman, politician, and served as a Union major general in the American Civil War. His ability to recruit German-speaking immigrants to the Union armies received the approval of President Abraham Lincoln.
wikipedia

Polish

Union General Włodzimierz Krzyżanowski who lead the Polish Legion

In Washington, D.C., he enlisted as a private two days after President Abraham Lincoln called for volunteers. He recruited a company of Polish immigrants which became one of the first companies of Union Soldiers. Krzyzanowski then moved his company to New York and enlisted more immigrants and soon became a Colonel of the 58th Infantry Division, listed in the official Army Register as the "Polish Legion." He participated in the Civil War battles of Cross-Keys, Bull Run, Chancellorsville, and Gettysburg. President Lincoln promoted him to General.
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After the war, he served as an administrator in the newly acquired territory of Alaska. He died on January 31, 1887, (and is buried at Arlington National Cemetery). Arlington National Cemetery

Confederate Colonel Gaspard Tochman who formed the Confederate Polish Brigade.

Gaspar Tochman (1797-1880), a major in the Polish army who participated in the failed 1830 revolt against Russia. Exiled, in 1837 he immigrated to the United States, where he practiced law, wrote, and lectured. During the Civil War he recruited the Polish Brigade (14th and 15th Louisiana regiments) of Jackson’s Corps. A colonel in the Confederate army, he sought unsuccessfully the rank of brigadier general. Fredericksburg Stafford Spotsylvania Historical Marker

Bermuda

First Sergeant Robert John Simmons

A Bermuda soldier who fought against the Confederacy in the American Civil War would probably have been one of the first black recipients of the US Medal of Honor had he not succumbed to his wounds following the pivotal Battle of Fort Wagner. Burmuda News

Italy

The Garibaldi Guard organized by Francesco Casale

The Garibaldi Guard was the nickname of the 39th New York Infantry, a regiment of Italian-Americans recruited mostly from New York City under the auspices of Francesco Casale and other Italian leaders in the North. Most of the members of this regiment were men who had fought under Giuseppe Garibaldi, the freedom fighter and republican agitator; they wore a distinctively styled red shirt as part of their uniform to show their connection to their countryman, whose partisans had worn such a shirt in Italy. Other Italian nationals joined the guard as well, however, out of a feeling that the Union's cause matched their own ideals of freedom and equal justice. They also viewed the Northern ideology as closely- allied with the aims of Garibaldi and felt such alliance lent credence to the great patriot's ideas, since they were clearly being adopted by other nations.
Civil War Home

To add to Twelfth's answer, Britain almost entered the American Civil War on several occasions, not just over the Trent Affair. Keeping Britain out of the war was an ongoing diplomatic struggle both Lincoln and his secretary of state William Seward were occupied with. This background task ultimately cast a shadow over the entire war as it was part of the inspiration for Lincoln changing the Unions motivation for conducting the war, the new goal being the eradication of slavery which most people today believe was why the Union fought the American Civil War.

Industrialized Britain had significant economic interests in the South. Namely Britain imported the cotton to feed it's industrialized looms and fuel it's economy, from the agricultural mass production institutions of the American South called plantations. Twelfth also mention that initially the stated reason the North invaded the South, in the beginning of the Civil War was to preserve the union. That is true, what is also true is when Lincoln made the abolition of slavery one of the Union's objectives in the war with one of the most famous executive orders, the Emancipation Proclamation Sept 1863, he had his eye on Britain.

As Lincoln had hoped, the Proclamation turned foreign popular opinion in favor of the Union by gaining the support of anti-slavery countries and countries that had already abolished slavery (especially the developed countries in Europe). This shift ended the Confederacy's hopes of gaining official recognition.
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The Emancipation Proclamation served to ease tensions with Europe over the North's conduct of the war, and combined with the recent failed Southern offensive at Antietam, to cut off any practical chance for the Confederacy to receive British support in the war Wikipedia