Where are the source and mouth of the Mississippi River located?
About 150 miles from where I am sitting at the moment is Lake Itasca, a Minnesota state park near the town of Bemidji, which is noted primarily for its exceedingly large statues of the mythological figures of Paul Bunyan and Babe the Blue Ox. Lake Itasca is the source of the Mississippi River. Two-thousand-three-hundred miles away is the Gulf of Mexico, which is the "mouth" of this great river. The mouth of the river is referred to as the Mississippi River Delta, and runs along the coast of Louisiana. A number of other large rivers flow into the Mississippi, most prominently, the Ohio and Missouri Rivers. [Note: there are several competing estimates of the river's length, with the Itasca State Park publishing the longest such estimate of 2,552 miles, and the U.S. Geologic Survey estimating it at 2,300 miles.]
The importance of the Mississippi River to American history cannot be overstated. The Mississippi River Delta region was colonized by the French starting in the late 17th century before being sold to the United States in 1803 as part of the Louisiana Purchase. To this day, the region, especially the City of New Orleans, retains much of its French character. Subsequent to the region's transition to the newly-established United States of America, the Mississippi River increased in importance, becoming integral to the development of the agricultural economy that dominated the South.
The source of the Mississippi River is at Lake Itasca in Minnesota, about 200 miles north of Minneapolis. At its source, the river is only about 20 feet wide and can easily be waded across. The mouth of the Mississippi is in Louisiana. There, near the town of Pilottown, the Mississippi flows out into the Gulf of Mexico.
The Mississippi is, of course, a very long river given that Minnesota is a very long way from Louisiana. Different lengths are given for the Mississippi River, but the length given in the link below is 2,350 miles. This does not include the length of the Missouri River, which rises in Montana and flows into the Mississippi a little north of St. Louis, Missouri.
So, the brief answer to your question is that the source of the Mississippi is at Lake Itasca and the mouth of the river is the Gulf of Mexico.