The tripartite of the nature of man can be seen both in the Old and New Testaments of the Scripture.
"And the Lord God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul" (Genesis 2:7, KJV).
Man was formed from the 'dust of the ground' that was the material part of man, which is the "Body". The 'breath of life' was next breathed into him, which didn't just contain life but also "Spirit" and he then became a living "Soul". The Spirit and Soul are the immaterial parts of him.
Now, animals: they are dichotomous, this means they just have life, which keeps their Body (not in God's image) alive in the physical and Spirit which of course gives them a certain kind of consciousness about living: eating, danger, recognition, but they certainly are not conscious about right and wrong. This is because they do not have a free will, they have no soul. Humans, on the other hand, are trichotomous, that is to say, like I said, they have body, spirit and soul. They have free will to make decisions, not just based on instincts like animals but based on right and wrong.
However, animals could possibly have a place they go when they die, though we weren't given full understanding about that (I guest it's not so important that's why the Bible didn't say much about it).
How about the New Testament? There are many verses actually where man's three natures appeared, but just to mention few, in Hebrew4:12,
"For the word of God is quick, and powerful, and sharper than any twoedged sword, piercing even to the dividing asunder of soul and spirit, and of the joints and marrow, and is a discerner of the thoughts and intents of the heart."
The Soul is mentioned here, the Spirit next, and of course the Body (which entails joints and marrow). Though three distinctive parts, yet the Word of God gets to reach them individually.
Now, I said, "Body, Spirit and Soul", but some would say, "Spirit, Soul and Body". What does matter is understanding how specially distinctive they are and being able to get how they are inter-related and how active in man as a being.
*WHO MADE THEM?
Let's see Revelation 5.13,
"And every creature which is in heaven, and on the earth, and under the earth, and such as are in the sea, and all that are in them, heard I saying, Blessing, and honour, and glory, and power, be unto him that sitteth upon the throne, and unto the Lamb for ever and ever."
Kindly, carefully read the above verse for the second time.
The spirits of all beings, I mean every single existing creature or entity in heaven, on earth, under the earth and in the sea perceives the existence of the One that sits on the throne and that rules over the affairs of all realms, provinces and territories of beings.
They are very much aware of who made them.
Those in the heavens (Angels and all inhabitants therein) cry, "Hosanna in the highest!" They render praises and service to that One on the throne. When one (Lucifer) thought he could change that, he was placed where he belongs, the ground.
Those on earth (Humans) understand that they didn't just happen by a random occurrence. They might not have all got to really know that One on the throne in terms of personal encounter or experience, but they do know that there is One who, of course, sits and rules over all. Which is why it has made so many ignorant ones deviced divers and wrong means of worshipping Him, some through idols others through some kind of human personality. The bottom line here is that they do know there's the mighty One who made everything and everyone. No wonder in the Bible they didn't know how else to find Him, as a result of their ignorance, so they decided to build a temple which they tagged, "TO THE UNKNOWN GOD". Unknown, in the sense that though they didn't recognize Him like Abraham did, like David and others did, but they did know that He existed and of course He still does exist.
How about those under the earth and in the sea? They are majorly animals and perhaps spirits, principalities, etc. They also heard the writer of Revelation saying, "Blessing, and honour, and glory, and power, be unto him that sitteth upon the throne, and unto the Lamb for ever and ever." Though they may not be as intelligent and thoughtful as man, yet their spirit acknowledges something or someone great when they behold it.