Descartes and the dualist intuition
The distinction feels obvious. You can hold a brain in your hand; you cannot hold a memory or a longing. In the seventeenth century Rene Descartes turned that intuition into a theory called substance dualism: mind and matter are fundamentally different kinds of stuff. The essence of matter is being extended in space, while the essence of mind is thinking, and the two, he claimed, could in principle exist apart.
Descartes's theory ran into trouble almost immediately, and the objection still bites. Princess Elisabeth of Bohemia asked him in 1643 how something with no physical location could push a physical body around at all. If the mind is not in space, what is its handle on the neurons? This interaction problem has no satisfying dualist answer, which is a large part of why most scientists and philosophers moved on.
The mainstream view: the mind is what the brain does
Today the standard scientific position is physicalism: mental states are ultimately states of the physical brain and body. The evidence is everywhere. A drug, a lesion, a lack of oxygen, or a magnetic pulse to the cortex changes thought, mood and perception in lawful, repeatable ways. Split the connection between the two hemispheres and you appear to split aspects of the mind. Minds and brains rise and fall together.
The most influential physicalist framework is functionalism, developed by thinkers including Hilary Putnam and Daniel Dennett. It says a mental state is defined not by what it is made of but by what it does, the causal role it plays between inputs, other mental states and behavior. On this view the mind relates to the brain roughly as a running program relates to the hardware, or as digestion relates to the stomach: not a separate thing, but the activity the organ performs.
Where the question stays open
Physicalism is the mainstream, but it should be held with intellectual honesty. Explaining how the brain processes information, controls behavior and reports on its own states is the tractable part, and it is going well. Explaining why all that processing is accompanied by felt experience, why there is something it is like to be the system, is the part that resists a clean physical account. That gap is what David Chalmers named the hard problem.
So the fair summary is this. The mind and the brain are not two objects sitting side by side; the best-supported view is that the mind is what the brain does. Dualism survives mainly because the felt quality of experience is hard to capture in physical terms, and a handful of serious philosophers still take that seriously. The everyday distinction is useful; the metaphysics behind it is not fully closed.