Judaism has always believed in reincarnation- its just the modern liberal sects that tend to discard it. The composition of the soul and how it relates to reincarnation is discussed in the Zohar, as well as in Likutei Amoraim (also known as Tanya).
Now the biggest issue that some people have with reincarnation often is: If we have multiple lives- how does everyone get resurrected in the time of mashiach? After all- which of those lives would come back?
So, to understand reincarnation and how it can be understood in light of techias meitim in the time of Mashiach- you need to understand the understanding of the soul.
In the simplest way of explaining it, the soul has three parts: Ruach which is tha animating spirit, think of it as the animal soul which every living being has and which dies when the person dies. Neshamah- this is the holy spark from G-d- it is 100% pure, cannot sin ad is sent into this world to be able to experience spiritual growth.
Now these two parts represent 100% physicality (Ruach) vs 100% spirituality (Neshamah)- and the two cannot combine or influence each other- which is where the third part comes in- the Nefesh. This si the "You" part- your intellect, mind, personality. It is the part that ties the spiritual to the physical and enable spiritual growth through the making of spiritual choices.
Now- when we die the Ruach dies- it plays no further part as its role was to animate the body. The Neshamah is 100% pure- it cannot sin and thus cannot be judged- it is the Nefesh that is judged- your actions, choices etc- and that which undergoes purification in gehinnom and then reward in the world to come. When we speak about reincarnation- we are referring to the fact that the Neshamah returns to this world to continue its spiritual growth. When we talk about Techias hameitim (Resurrection of the dead) in the time of Mashiach- we refer to the fact that the Nefesh is given a new body and thus returns to earth. Thus since there is an unique instance of the Nefesh for each time the Neshama is reincarnated, there is an unique person to be resurrected each time.
Orthodox Jew; Zohar; Likutei amoraim