I am quoting here from an article I wrote, hope this helps.
Who is the Pitr?
Pitr basically refers to departed forefathers. Hinduism does not
accept the theory of permanent heaven or hell. The Hindu scriptures
speak about how Karma determines the journey of a particular
individual after the death of his body and how the individual will
eventually take a re-birth in a new body.
The scriptures speak about three paths a departed soul may take based on its actions when it was alive. Some who have lived life
according to dharma and have practiced extensive devotion and
meditation, then by the merit of such actions, he or she will attain
Devaloka (the realm of the gods) through the northern path.
Some people who have practiced various duties and rites alone without devotion or meditation, may have to take a southern path to
“Pitrloka” (the realm of ancestors/manes). On the other hand, those
who live life selfishly and perform adharmic actions, may have to
enter realms of hells called Naraka and suffer there for their
unrighteous actions.
But, ultimately, all of them, according to their actions and its
fruits, will again take birth as humans in the physical universe.
Without going into further details, it is suffice to say that, once a
person dies, he discards his gross body and travels to Pitrloka in his
subtle body and stays there until his karmic fruits that took him
there get exhausted.
Manu Smriti (3.192) describes the Pitr as “free from anger, careful of
purity, ever chaste, averse from strife, and endowed with great
virtues.” Our forefathers who managed to reach there did so based on
the merits of their own actions as well as due to the efforts of their
offsprings’ who performed proper funeral rituals.
But, Pitrs, similar to humans, have unfulfilled desires. Many people
when they die, die in misery and frustration. Most of them have
strong, unfulfilled desires. These unfulfilled desires get carried
even after one discards the physical body, as the desires belong to
the mind, which is associated with the subtle body.
Therefore, Pitrs refer to our immediate parents who have departed, to
our forefathers with whom we share a blood relationship, and in
general to all ancestors who have come before us.
Why observe Pitr-Paksha?
As noted above, Pitrs refers to our ancestors. No person can take
birth in this physical universe without being given birth by his/her
parents. Therefore, each person owes to a debt to his parents for
giving him/her an opportunity to enter this Universe. Then, the
parents support their children, give them education, and help them
grow up. Even though all these are done out of love by the parents, a
Karmic bond and a debt is formed that the children owe to their
parents.
This Karmic debt is called as “Rina” and such a bond of debt between
two individuals is called as “Rina Bandha.” Now, it is not just
parents who are responsible for the birth of an individual. The
parents themselves were given birth by their parents. Hence, a person
has a Rina Bandha with the whole family and lineage into which he or
she is born.
The famous King Bhagiratha did practice severe Tapasya (austerity) to
bring River Ganga on earth so that he could free his ancestors from
Sage Kapila’s curse. This was his way of paying the debt to his
ancestors.
In modern scientific language, a person derives his genes that are the
basis of his existence from his ancestors. Therefore, there is a debt
even on that count.
It is said that during the period of Pitr-Paksha, the Pitrs come down
from their abodes in Pitrloka and reside near their descendants. If,
one were to offer food and water and worship the Pitrs during this
time, they will be very pleased and they will in-turn bless the
descendants.
Manu Smriti (3.259) says that after performing Shraddha (a ritual to
satiate Pitrs), the performer must pray for the blessings of Pitrs
thus: “May liberal men abound with us! May (our knowledge of) the
Vedas and (our) progeny increase! May faith not forsake us! May we
have much to give (to the needy).”
P V R Narasimha who has prepared simple manuals for worshipping Pitrs says: “By thinking of the deceased ancestors with gratitude and
trying to give them an emancipation, one is actually trying to free
oneself from various Karmic predispositions that one has as a result
of the Rina (Karmik debt) with several people. One can view this as an
external event of satisfying and emancipating an external entity (a
Pitr). Alternatively, one can view this as an internal event of
satisfying and releasing an internal Karmik predisposition.”
Thus, the worship of Pitrs during Pitr-Paksha, will not only help a
person to pay back his Karmic debts to his ancestors by satiating
their desires, but also helps to gain their blessings that will give
rise to both material and spiritual welfare. Most importantly, the
worship of Pitrs will help one to purify one’s mind by reducing the
internal Karmic burden.