First we have to define what is "soul". I take it as meaning "a continuity of the same permanent self that exists after death".
The three characteristics of all phenomena or existence are impermanence, unsatisfactoriness and insubstantiality, which I had mentioned before. The third characteristic is also explained as "no-soul". The Pali term is "anatta"..."no self". The Buddha's teachings stand aloft in this concept of "anatta".
A physical being is composed of the physical body and consciousness which is a form of energy. This "etheral energy" gives live to form. Without this energy the form is dead. However when this physical form is worn out and fails to function, the etheral energy takes on another form to start a new cycle. This process is called rebirth.
"Rebith" takes on a totally different perspective. It views existence as an on-going process, just like the electric current lighting up a bulb. The "bulb" is like the physical body. When this bulb is blown, the current is still there and when a new bulb is fixed on, it lights up again. So you can see that there is a continuity in the electric current, but the current flowing through is not the same at any one moment. Our existence is in this state of flux. When the body is dead, the life energy seeks another form to "reborn". The type of rebirth will depend on the nature of this store-house of life energy. If this store-house of energy has more wholesome characteristics, then it will seek rebirth in a more conducive environment. On the other hand, if it has more negative characteristics, then its rebirth will be in a more unwholesome environment. As the life force is always in a flux, there is no permanent and unchanging soul, but a changing personality that exists throughout. To describe this process of rebirth, the Buddha used the term "Not exactly the same, yet not totally different". Another example will be a lighted candle. It is not the same light that we see, but not totally different.