South of (roughly) the 39th parallel -- the Mason-Dixon line. In antebellum times, before the westward expansion, that line was the unofficial division between North and South, also between the slave and free states.
As important as it is to define our terms, it is of prime importance to make sure that our defining criteria covers all the bases. A big part of what defines North and South is how the people at the time viewed themselves and their respective states. For example, Delaware, a slave state, and a southern state in relation to the Mason-Dixon line, traditionally viewed itself as part of the North. Delaware refused to secede and sent most of its men to the army of the North.
As the country expanded to the west, large chunks of what we now think of as the Midwestern states -- Ohio, Indiana and Illinois -- were south of the Mason-Dixon line, but were culturally, economically and politically aligned more with the North than the South. However, the extreme southern counties of those states had somewhat stronger ties with the South.
The people in the midwest considered themselves to be "Westerners", despite the fact that they lay to the east of the Mississippi River. With the expansion of the country across the continent to the Pacific, they came to be known as Midwesterners. The Confederate government referred to the war from Tennessee and Alabama to the Mississippi River as the "Western Theater" of the war. Indeed, the military command in that region was first called the Department of the West.
Arkansas and most of Louisiana lay to the west of the Mississippi River, but have always considered themselves to be part of the South, not part of the West. Texas is kind of in a league by itself -- very Southern in the southeastern counties, westward looking (at least) in the western and northern counties.
Maryland, Kentucky and Missouri -- the so-called Border States -- were in a more ambiguous position. They had firm cultural, political and economic ties to the South and a sizeable pro-secession population. One of the earliest campaigns of the war involved the occupation of those states by Federal troops to ensure their retention in the Union. All three states provided regiments to both sides in the war.
Finally, it is important to keep in mind that the populations of the states further to the west consisted to a great degree of immigrants from the eastern states. So, while we may say they were western states according to their location on the map, they tended to look upon themselves as southerns or northerners, depending on where their families had come from.
Therefore, I don't know that it's particularly helpful for us modern-day types to be too definitive in issues like this. The war was fought by people, not states, and we should go along with how those people defined themselves.