In the Ramayana war, Rama killed Ravana with a weapon given to him by sage Vasishtha. What was the name of the weapon?
Sri Rama killed Ravana by using the weapon presided over by brahma, which was given by Sage Agastya. Please check the verses from verses 2 onwards sarga 108, Yuddha Kanda of Valmiki Ramayana.
Check the Verse 3 from the link http://www.valmikiramayan.net/utf8/yuddha/sarga108/yuddha_108_frame.htm
Then, the valiant Rama, who was reminded thus by Matali, took hold of a blazing arrow, which was given by Brahma and which in turn was given to him by the glorious sage, Agastya earlier in the battle-field and which looked like a hissing serpent. (Canto-6; Sarga-108; Verse 3)
Rama did not kill Ravana with Vasishta's weapon.. As I discuss in this answer, Rama killed Ravana with a Brahmastra, and it was given to him by the sage Agastya, not his guru Vasishta. Here is how the Yuddha Kanda of the Ramayana describes it:
Then, the valiant Rama, who was reminded thus by Matali, took hold of a blazing arrow, which was given by Brahma and which in turn was given to him by the glorious sage, Agastya earlier in the battle-field and which looked like a hissing serpent. Having been made formerly for Indra, the lord of celestials by Brahma, the lord of creation of infinite strength, it was bestowed in the past on the ruler of gods, who was desirous of conquering the three worlds....
Making it sacred by a special formula as per the procedure specified in scriptures, Rama who was endowed with an extraordinary strength then fixed that arrow which was the foremost among the three worlds, capable of removing the fear of Ikshwaku dynasty, taking away the glory of the enemies and bestowing joy to one's own self on his bow.
While that excellent arrow was being fixed by Rama, all the beings were frightened and the earth trembled. That enraged Rama, stretching his bow well and with an attentive mind, hurled that arrow which can tear off the vitals, towards Ravana. That arrow, which was inviolable as a thunderbolt hurled by the arms of Indra and irresistible as Yama the lord of Death, fell upon Ravana's chest. That arrow, released with great speed and which was capable of destroying the body, tore off the heart of that evil-minded Ravana. That arrow, which was capable of causing death to the body, after taking away the life of Ravana and having been anointed with blood, penetrated the earth.... Having lost his life, that king of demons for his part, who was endowed with terrible swiftness and invested with great splendour, fell down from the chariot to the ground, like Vritra the demon when the latter was struck down by the thunderbolt.
That line at the end, by the way, is an allusion to story of Indra killing Veitrasura, which I discuss here.
The answer is present in Valmiki's Ramayana
Mathili tells to Lord Rama
O, Lord! You can employ a mystic missile presided over by Brahma the lord of creation. The time for his destruction has come now, as expressed by the celestials.
Then, the valiant Rama, who was reminded thus by Matali, took hold of a blazing arrow, which was given by Brahma and which in turn was given to him by the glorious sage, Agastya earlier in the battle-field and which looked like a hissing serpent. .. That enraged Rama, stretching his bow well and with an attentive mind, hurled that arrow which can tear off the vitals, towards Ravana. That arrow, which was inviolable as a thunderbolt hurled by the arms of Indra and irresistible as Yama the lord of Death, fell upon Ravana's chest.
BrahmAstra, which blazed RAvana & his chariot with fires. This also could be the reason of "Dussera" being celebrated in certain way.
And beholding that terrible form of illusion displayed by the Ten-necked Rakshasa, the monkeys fled in fear in all directions. Then the descendant of Kakutstha, taking out of his quiver an excellent arrow furnished with handsome wings and golden feathers and a bright and beautiful head, fixed it on the bow with Brahmastra mantra. And beholding that excellent arrow transformed by Rama, with proper mantras into a Brahma weapon, the celestials and the Gandharvas with Indra at their head, began to rejoice. And the gods and the Danavas and the Kinnaras were led by the display of that Brahma weapon to regard the life of their Rakshasa foe almost closed. Then Rama shot that terrible weapon of unrivalled energy, destined to compass Ravana's death, and resembling the curse of a Brahmana on the point of utterance. And as soon, O Bharata, as that arrow was shot by Rama from his bow drawn to a circle, the Rakshasa king with his chariot and charioteer and horses blazed up, surrounded on all sides by a terrific fire. And beholding Ravana slain by Rama of famous achievements, the celestials, with the Gandharvas and the Charanas, rejoiced exceedingly. [MahAbhArata, Vana Parva -- as said by sage Markendeya]
The weapon which was used to kill Ravan was a weapon given by Brahma. It is not presided by Brahma. I am directly quoting the answer of a friend of mine on Quora.
Rama was mulling that the weapons that destroyed Mareech, Khara, Dushan and Vali were effete against Ravan (1)
Rama used Brahmastra to kill Mareecha (2) and Khara (3).
Clearly, the Brahmastra was effete against Ravan. Matali urged Rama to use the mystic missile given to him through Agastya by Brahma. (4)
Sita reported Hanuman that she hopes to see Ravan along with his thugs by the terrible weapon of Lord Rudra.(5)
raudreNa kaccit astreNa raameNa nihatam raNe | drakShyaami alpena kaalena raavaNam sasuhR^ij janam || 5-36-27
"Within a short time, I hope see Ravana killed along with his companions in the battle-field by terrific weapons of Rudra burning brightly."
Look likes a Pashupata weapon to me.
Lord Rama used various astras against Ravana and he too bombared many in return. Some astras Lord Rama used are over here:
Vasavi Shakti Astra; Brahma Astra; Bramhashira Astra; Varuna Astra; Agneyastra; Brahmanda Astra; Surya Astra; Jyothiksha Astra; Maheshwara Astra; Linga Astra; Shiva Astra; Sudharshana Astra; Venkatesa Astra; Narayanan Astra; Karma Astra; Bhumi Astra; Vayva Astra; Trimurti Astra; Garuda Astra; Bhakti Astra; Siddhi Astra; Kali Astra; Durga Astra; Yama Astra; Indra Astra; Chandra Astra; Asvini Astra; Rishi Astra; Gandharva Astra; Maya Astra; Deva Astra; Yaksha Astra; Parvata Astra;