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.