Naval combat was not as frequent as clashes between land troops, but it did happen, and navies played a crucial role in the war. You could break down Civil War naval operations into two basic kinds: operations relating to blockades along the coast (either maintaining or trying to break through), and riverine/amphibious operations, where naval vessels worked in tandem with land forces to take a strategic point. These two kinds of operations worked together toward a greater objective, discussed below.
At the outset of the war, the US implemented a strategic blockade. This was part of a strategy known as the Anaconda Plan, developed by US General Winfield Scott. It called for a blockade of the entire Confederate coastline and control of the Mississippi River, in order to completely cut off the poorly-industrialized South from foreign arms and ammunition. Maintaining this blockade necessarily took the overwhelming majority of the US Navy, which had inadequate numbers in the early months of the war. The kinds of ships that blockades the coasts were a mix of sail and steam powered vessels primarily made of wood and usually of a frigate class (think 250-300 ft long, ~3000 tons), and well-armed. Some were converted from civilian vessels in the haste to boost the number of ships available.
For their part, the Confederates' never had much of a navy, and so they tried a number of solutions. The first was to make ironclads, which are exactly what they sound like: up-armored ships. Their first ironclad, CSS Virginia, was built on the hull of the scuttled USS Merrimack. In early 1862 it led a squadron of five other Confederate ships on the Battle of Hampton Roads. The first day of the Battle was almost comically one-sided: The Confederates suffered 2 casualties and lost no ships, compared to the Union's 250 casualties, 2 ships sunk, and 3 aground. The second day was more of a fight, since the US Navy had also been working on an ironclad, the USS Monitor. The Monitor had a rotating turret with two dahlgren cannons and barely any of its hull was above water, making it nimble and a small target. The Virginia and the Monitor dueled for a few hours, but neither one prevailed before they broke off the engagement. This was the first ironclad vs ironclad battle in world history.
Another solution the Confederates tried was, I kid you not, submarines. They created an absolute deathtrap of a machine called the HL Hunley. It sank twice during test runs, killing 5 crew the first time and all 8 the second time, though it was returned to service each time. It was used in Charleston harbor against the US naval blockade, and it actually managed to sink a ship with its spar-mounted torpedo. This was the first successful submarine attack in world history. Sadly, the explosion also damaged the Hunley, and it sank a third time, killing all hands aboard.
Other engagements took place on the Mississippi. The Vicksburg campaign is a prime example of this. Gunboats on the Mississippi worked with Grant's army, providing them with transportation and supply (as they did when they ran past the battery at Vicksburg, essentially bypassing it) and were also used to protect the army when it was cornered at Shiloh. Other similar operations include the Battle of New Orleans in 1862 and the Battle of Mobile Bay in 1864. Both times, naval forces coordinated with ground troops in amphibious operations to take out major ports.
Sadly, not many ships survived the war to the present. The only one that is still intact and on display that I mentioned is the Hunley. The Virginia was scuttled after Norfolk fell to the Union and the Monitor sank off Cape Hatteras in a storm during the war, though its turret has been recovered.