The concept, whether you call it rebirth or reincarnation or what have you, is as described in the suttas, especially those about dependent origination. The Nidanasamyutta (SN 12) is all about dependent origination and has many suttas. Here is SN 12.2 which gives plain definitions of each link of dependent origination.
At Savatthī. “Bhikkhus, I will teach you dependent origination and I will analyse it for you. Listen to that and attend closely, I will speak.”
“Yes, venerable sir,” those bhikkhus replied. The Blessed One said this:
“And what, bhikkhus, is dependent origination? With ignorance as condition, volitional formations come to be; with volitional formations, consciousness … as in preceding sutta … Such is the origin of this whole mass of suffering.
“And what, bhikkhus, is aging-and-death? The aging of the various beings in the various orders of beings, their growing old, brokenness of teeth, greyness of hair, wrinkling of skin, decline of vitality, degeneration of the faculties: this is called aging. The passing away of the various beings from the various orders of beings, their perishing, breakup, disappearance, mortality, death, completion of time, the breakup of the aggregates, the laying down of the carcass: this is called death. Thus this aging and this death are together called aging-and-death.
“And what, bhikkhus, is birth? The birth of the various beings into the various orders of beings, their being born, descent into the womb, production, the manifestation of the aggregates, the obtaining of the sense bases. This is called birth.
“And what, bhikkhus, is existence? There are these three kinds of existence: sense-sphere existence, form-sphere existence, formless-sphere existence. This is called existence.
“And what, bhikkhus, is clinging? There are these four kinds of clinging: clinging to sensual pleasures, clinging to views, clinging to rules and vows, clinging to a doctrine of self. This is called clinging.
“And what, bhikkhus, is craving? There are these six classes of craving: craving for forms, craving for sounds, craving for odours, craving for tastes, craving for tactile objects, craving for mental phenomena. This is called craving.
“And what, bhikkhus, is feeling? There are these six classes of feeling: feeling born of eye-contact, feeling born of ear-contact, feeling born of nose-contact, feeling born of tongue-contact, feeling born of body-contact, feeling born of mind-contact. This is called feeling.
“And what, bhikkhus, is contact? There are these six classes of contact: eye-contact, ear-contact, nose-contact, tongue-contact, body-contact, mind-contact. This is called contact.
“And what, bhikkhus, are the six sense bases? The eye base, the ear base, the nose base, the tongue base, the body base, the mind base. These are called the six sense bases.
“And what, bhikkhus, is name-and-form? Feeling, perception, volition, contact, attention: this is called name. The four great elements and the form derived from the four great elements: this is called form. Thus this name and this form are together called name-and-form.
“And what, bhikkhus, is consciousness? There are these six classes of consciousness: eye-consciousness, ear-consciousness, nose-consciousness, tongue-consciousness, body-consciousness, mind-consciousness. This is called consciousness.
“And what, bhikkhus, are the volitional formations? There are these three kinds of volitional formations: the bodily volitional formation, the verbal volitional formation, the mental volitional formation. These are called the volitional formations.
“And what, bhikkhus, is ignorance? Not knowing suffering, not knowing the origin of suffering, not knowing the cessation of suffering, not knowing the way leading to the cessation of suffering. This is called ignorance.
“Thus, bhikkhus, with ignorance as condition, volitional formations come to be; with volitional formations as condition, consciousness…. Such is the origin of this whole mass of suffering. But with the remainderless fading away and cessation of ignorance comes cessation of volitional formations; with the cessation of volitional formations, cessation of consciousness…. Such is the cessation of this whole mass of suffering.”
The “various orders of beings” are beings of the 3 spheres of existence, beings of the: sense, form, and formless spheres.