This is the first Battle of Memphis. This one was fought on the water. The water was the Mississippi River right near Memphis. This was the start of the Union's drive to gain control of the Mississippi River and cut the Confederacy in half. A major goal of the Union. Eventually the idea was to capture New Orleans one of the major ports in the Confederacy.
Here's some more information on the battle:
After the Confederate River Defense Fleet, commanded by Capt. James E. Montgomery and Brig. Gen. M. Jeff Thompson (Missouri State Guard), bested the Union ironclads at Plum Run Bend, Tennessee, on May 10, 1862, they retired to Memphis. Confederate Gen. P.G.T. Beauregard ordered troops out of Fort Pillow and Memphis on June 4, after learning of Union Maj. Gen. Henry W. Halleck's occupation of Corinth, Mississippi. Thompson's few troops, camped outside Memphis, and Montgomery's fleet were the only force available to meet the Union naval threat to the city. From Island No. 45, just north of Memphis, Flag-Officer Charles H. Davis and Col. Charles Ellet launched a naval attack on Memphis after 4:00 am on June 6. Arriving off Memphis about 5:30 am, the battle began. In the hour and a half battle, the Union boats sank or captured all but one of the Confederate vessels; General Van Dorn escaped. Immediately following the battle, Col. Ellet's son, Medical Cadet Charles Ellet, Jr., met the mayor of Memphis and raised the Union colors over the courthouse. Later, Flag-Officer Davis officially received the surrender of the city from the mayor. The Indiana Brigade, commanded by Col. G.N. Fitch, then occupied the city. Memphis, an important commercial and economic center on the Mississippi River, had fallen, opening another section of the Mississippi River to Union shipping.
The city of Memphis allowed the Union to control the Mississippi River down to Vicksburg where one of the major and most important battles of the Civil War would take place in June of 1863.