What is the 'spirit' of a person, and are the soul and spirit the same thing?

They are basically the same. The Greek word for soul, psyche, has multiple meanings; it can refer to the physical person, as when it says there were eight souls on the ark, meaning eight human beings, and it has about a half-dozen derivatives, including a carnal aspect. We all have a spirit, a spiritual essence, which we usually call our soul, and then the body, the physical part. When your body goes into the grave, your spirit leaves it and goes to its designated place, heaven, hell, or purgatory; there is no soul sleep. But when spirit and body are integrated as one living human being, you cannot tell them apart, and we assign words to that integration, one of which is soul, so you can say the soul is with the body or with the spirit depending on context. It is like the word heart, which is anatomically an organ that pumps blood but which we use for the centralized, vital center of the person, or mind, which is the intellect composed of spirit and body. When Jesus says love me with your whole heart, mind, soul, and strength, he wants every piece of you, because we are this strange combination of spirit and body.