I have compiled this answer after consulting two books that i have namely BenimAdhava's Bengali Panchanga' and a prayoga book called Brihat Visuddha Nitya Karma Puja Paddhati (this book has all the details of how to do the Vrata Puja & from the panchanga book's description of Vratas chapter i took the Vrata katha and other information).
Who is Goddess VipattArini?
Vipad means dangers, calamities. TArana from TrAna
means to protect or to save. So, VipattArini is the Goddess who protects devotees from dangers and calamities.
Godddess VipattArini is actually Goddess DurgA. This can be confirmed from VipattArini Puja mantras as well as the Sankalpa mantra given below.
The main mantras used in the PujA are
Durge durge rakshani swAha or Hrim vipattArini durgA devyai namaha
and which prove that the Gooddess we are worshiping as VipattArini is Goddess Durga herself.
Also, the Avahana mantra viz:
Om hrim VipattArini durgA devi ihA gaccha ihA gaccha etc
proves the same.
The DhyAna Sloka used in the PujA also proves that it is Goddess DurgA's pujA.
KAlAbhrAbhAm... DhyAyedurgAm JayAkhyAm tidasaparivritAm sevitAm
sevitAm siddhikAmyai.
Also, from the sankalpa mantras used in this vrata we find:
..... YAvAdvipannAshapurvak sasukhAvaidhavyakAmA SrivipattArini
DurgApritikAmA..Sri VipattArini vrataham karshyAmi.
....
.. For destroying all the dangers and calamities that are looming
large on me, for the increment in my happiness , for preventing widowhood
(for women) , for pleasing Goddess DurgA, i am performing Sri
VipattArini Vrata..
And, this sankalpa mantra also depicts the fruits that one achieves upon completing the vrata.
When to perform this vrata?
The vrata is to be done in the Hindu AshAda month (which ended just
few days back) And, the day has to be either tuesday or
saturday and it has to be on a day that falls between sukla tritiya to suklA navami tithi (sukla is bright phase of the moon) of
the aforementioned month.
Upakarana or ingredients required
Ghata (kalas), mango leaves, coconut, one set of naivedyam, 13 types
of flowers (the number 13 is important in this Vrata), 13 types of
fruits each sliced into two halves, 13 types of whole fruits, 13 red
mauli threads, 13 durvAs , 13 beetle leaves and 13 beetle nuts.
Additional rules to be followed
On the day before the vrata, day one should consume only havishaya anna
(vegetarian food without onion, garlic etc). On the vrata day after
the completion of the puja, the vrata katha has to recited and
listened to by sitting on an asana. After that is done, the fast
must be broken while still sitting on the same asana. The foods
that are recommended to be consumed to break the fast are fruits, sweetmeats, and/or luchi (puri). Thirteen knots are made in the the
red mauli threads (with each knot carrying a durva ) and are then
worn by the males on their right hand and females on their left.
The Puja of Goddess Durga has to be done by appointed Brahmin priests
following the prescribed rules like GhatasthApana etc. And, as per one's
capability, charity and fees are to be duly given to the priests for
successful completion of the vrata.
The Vrata katha
This story, which i have translated from Bengali, has to be listened before breaking the fast.
In ancient times, in the Vidarva kingdom, there was a king whose wife
(the queen) was a great devotee of Mother Goddess and was a very
faithful wife to her husband too.
The queen sincerely took the VipattArni vrata and completed it. At
that time she also befriended a muchini (a low caste woman, the wife
of a cobbler).
Almost regularly the queen used to gift the muchini with many
eatables, fruits etc.
The muchini was terribly poor but upon receiving daily gifts from the
queen, the desire of gifting something to the queen in return also
grew big in her.
And, when she expressed her desire to the queen , the queen said "Okay
i will demand it from you someday". But many days elapsed still the
queen did not demand anything.
Finally, on one day, the queen said to the muchini " I have heard that
you people cook a meat (that is forbidden for the Hindus to consume) very well. Why
don't you bring some for me. Let me also see how good that meat actually
is?"
The muchini was very happy thinking that finally she can gift something to the
queen.
She, thus, cooked the forbidden meat with great care and brought it to the
palace after having covered it with some cloth. The queen will obviously not
taste the meat but will only see it ( this the muchini thought in
her mind).
Although the muchini took great care to hide the meat but servants in
the palace somehow still saw it and brought the incident to the king's
notice.
The king was furious as a result and said to the queen " Show me what that muchini
brought for you. If it is something forbidden (the king already heard
what it was from his servants) then i will cut your head off (as a
punishment).
The queen got pale in fear and started crying and praying to Goddess
VipattArini Durga.
Unable to bear the devotee's pain and distress, Goddess came and whispered in the
queen's ears " Dekhoge sob phool hoi gece" ( Go and see, all that
[forbidden meat] has been converted to flowers.
The queen then brought those flowers and showed them to the king. The
king reprimanded his servants and the queen's danger was thereby
averted. Thus the greatness of VipattArini spread among the masses and
the vrata became widely known.
Hope this helps.