Your soul is who you are; the essence of your being. Your spirit is given to connect with God. When the soul is used in the Bible, it can refer to the whole person: in its most basic sense, the word soul means "life".
As Paul explained, we are triune beings made up of body, spirit and soul. A good analogy would be the three temple courts: outer, inner, and the Holy of Holies. The outer court represents your body, the inner court represents your spirit, and the center of man, the soul, represented by the Holy of Holies.
I THESSALONIANS 5:23 And the very God of peace sanctify you wholly;
and I pray God your whole spirit and soul and body be preserved
blameless unto the coming of our Lord Jesus Christ.
Some scripture references to put things into context:
Genesis 2:7 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
Genesis 35:18 And it came to pass, as her soul was in departing, (for
she died) that she called his name Benoni: but his father called him
I Corinthians 2:11-12 For what man knoweth the things of a man, save
the spirit of man which is in him? even so the things of God knoweth
no man, but the Spirit of God. Now we have received, not the spirit of
the world, but the spirit which is of God; that we might know the
things that are freely given to us of God.
Ezekiel 36 explains the born again experience. First, God gives you a new heart to believe him. Then a new spirit that cleanses you and sets you aside for service. Finally, He puts His spirit in you.
Ezekiel 36:26-27 A new heart also will I give you, and a new
spirit will I put within you: and I will take away the stony heart
out of your flesh, and I will give you an heart of flesh. And I will
put my spirit within you, and cause you to walk in my statutes,
and ye shall keep my judgments, and do them.
The born again experience is also mirrored in the Old Testament temple worship. First, you approach God through the outer court, where the Word was taught (under Solomon's Porch). Next, you enter into fellowship with God in the inner court, by offering sacrifices and through worship with God's people. Finally, the veil having been rent by our Christ's sacrifice, we are able to enter into the Holy of Holies as "kings and priests" (Revelation 1:6) and enter into true fellowship with God.
Viewpoints on the Triune Nature of Man
Early 1900 teachers such as Lehman Strauss and William Branham taught that the body consisted of 5 natural senses (sight, taste, etc), and the middle realm (spirit) consisted of 5 senses (memory, reason, consciousness, affection, imagination).
Lehman Strauss taught that the center realm (soul) consisted of an additional five senses: faith, hope, reverence, prayer and worship.
William Branham on the other hand, taught that the center realm consisted of only one sense, either faith or doubt, depending on the person's state of salvation (if you had been filled with the Holy Ghost, you believed God's Word).
William Branham's viewpoint is consistent with the "floor plan" of Solomon and Herod's temples.
From Alfred Edersheim's "Sketches of Jewish Social Life in the Days of Christ":
The priests kept watch in the Temple in three places: in the house
Avtinas, and in the house Nitsuts, and in the house of Moked; and the
Levites in twenty-one places: 5 at the five gates leading into the
Temple (the Mountain of the House), 4 in the four angles within, 5 at
the five gates of the court, 4 in its four angles without, and 1 in
the chamber of offering, and 1 in the chamber of the vail, and 1
behind the Most Holy Place (the House of Atonement).
There were obviously more than just five gates into each court of the temple complex. But the order of approach, and the symbolism, was maintained by the way that the Levites guarded the gates. They guarded five gates of the body, five gates of the spirit, and one gate into the Holy of Holies (soul).