Question: "Do we have two or three parts? Are we body, soul, and spirit - or - body, soul-spirit?"
Answer: Genesis 1:26-27 declares, "And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth. So God created man in His own image, in the image of God created He him; male and female created He them."
These verses indicate that there is something distinct about humanity from all the other creations. Human beings were intended to have a relationship with God, and as such, God created us with both material and immaterial aspects. The material aspects are obviously those that are tangible: the physical body, organs, etc. and are considered to only exist as long as the person is alive. The immaterial aspects are those which are intangible: soul, spirit, intellect, will, conscience, etc. These characteristics are considered to exist beyond the physical lifespan of the individual.
All human beings possess both material and immaterial characteristics in their existence. It is clear that all mankind has a body, containing flesh, blood, bones, organs and cells. However, it is the intangible qualities of mankind that are often debated. What does Scripture say about these? Genesis 2:7 - states that Man was created as a living soul. Numbers 16:22 - And they fell upon their faces, and said, O God, the God of the spirits of all flesh, shall one man sin, and wilt thou be wroth with all the congregation? This verse names God as the God of the spirits that are possessed by all Mankind. Proverbs 4:23 - Keep thy heart with all diligence; for out of it are the issues of life. This verse indicates that the heart is central to Mankind’s will and emotions. Acts 23:1 - And Paul, earnestly beholding the council, said, Men and brethren, I have lived in all good conscience before God until this day. Romans 12:1 - I beseech you therefore, brethren, by the mercies of God, that ye present your bodies a living sacrifice, holy, acceptable unto God, which is your reasonable service. And be not conformed to this world: but be ye transformed by the renewing of your mind, that ye may prove what is that good, and acceptable, and perfect, will of God. It can be seen that there are various aspects of the immaterial part of humanity, and that all human beings share both material and immaterial qualities. This list of references just scratches the surface.
So, while much of the discussion of the immaterial aspect of humanity focuses on the soul and spirit, Scripture outlines far more than those two. Somehow, the aspects mentioned above (soul, spirit, heart, conscience, and mind) are connected and interrelated. The soul and spirit, though, definitely are the primary immaterial aspects of humanity. They likely comprise the other aspects. With this is mind, is humanity dichotomous (cut in two, body / soul-spirit), or trichotomous (cut in three, body / soul / spirit). It is impossible to be dogmatic. There are good arguments for both views. A key verse is Hebrews 4:12, "For the word of God is living and active and sharper than any two-edged sword, and piercing as far as the division of soul and spirit, of both joints and marrow, and able to judge the thoughts and intentions of the heart." This Scripture tells us at least two thing about this debate. (1) The soul and spirit can be divided. (2) The division of soul and spirit is something that only the Word of God can discern. We can be sure that as human beings, we possess a body, a soul, a spirit, and much more! However, rather than focusing on these aspects, it is better to focus on the Creator, of Whom we are "fearfully and wonderfully made" (Psalm 139:14).