अश्वत्थामन्, Aśvatthāman means the horse-voiced.
At the time of his birth, he neighed like the (celestial) steed Ucchaihsravas.
Drona, obedient to the injunctions of his father and moved by the
desire of offspring married Kripi, the daughter of Saradwat. And this
woman, ever engaged in virtuous acts and the Agnihotra, and the
austerest of penances, obtained a son named Aswatthaman. And as soon
as Aswatthaman was born, he neighed like the (celestial) steed
Ucchaihsravas.
Hearing that cry, an invisible being in the skies said,
'The voice of this child hath, like the neighing of a horse, been
audible all around. The child shall, therefore, be known by the name
of Aswatthaman, (the horse-voiced).
Drona used to instruct his own son in several superior methods (of using weapons), by devising a plan to engage the Pandavas, Kauravas and other disciples. However, Arjuna being very sharp, understood his Guru's methods, overcome those hurdles and learnt all the weapons that Drona taught to Aswatthaman.
Aswatthaman excelled everyone (in the mysteries of the science of arms). Arjuna, however, outdistanced everyone in every respect--in intelligence, resourcefulness, strength and perseverance.
Accomplished in all weapons, Arjuna became the foremost of even the foremost of car-warriors; and his fame spread all over the earth to the verge of the sea. And although the instruction was the same, the mighty Arjuna excelled all (the princes in lightness of hand). Indeed, in weapons as in devotion to his preceptor, he became the foremost of them all. And amongst all the princes, Arjuna alone became an Atiratha (a car-warrior capable of fighting at one time with sixty thousand foes).
Drona considered Arjuna to be the foremost of all his pupils, and became highly pleased and gave him very superior and irresistible weapon called Brahmasira with the methods of hurling and recalling it.
Aswatthaman might have got envious with Arjuna at this juncture. He forced his father to teach him the usage of that superior weapon Brahmasira. Out of fondness, he taught it to his son, which scriptures prohibit.
His father Drona remained loyal and lived Hastinapura. The Northern province of Panchala, which Arjuna won for Drona from Drupada, was ruled by Aswatthaman.
The greatest blunder that Drona did, after getting revenge against Drupada done, was to continue to live among kshatriyas, leaving his basic duties as brAhmana. This lead to development of rAjasic tendency within him, which finally led to devising a devilish plan to eliminate Abhimanyu, who was all alone.
Even Aswatthaman, who dutifully followed his father, became engrossed with cruelty, which ultimately led him to kill the sleeping army of Pandavas, after the fall of Duryodhana, in the night.
Though scriptures prohibit a Guru to impart knowledge of superior weapons to ineligible one, Still Drona taught Aswatthaman the usage of brahmasira, but did not teach him to withdraw it. Aswatthaman used it for the destruction of Pandava race.
Blinded by anger he allowed his brahmasira weapon, which he could not withdraw, to fall on the wombs of the Pandava women.
For this devilish act, Sri Krishna cursed Aswatthaman as follows:
For 3,000 years thou shalt wander over this earth, without a companion
and without being able to talk with anyone.
To conclude, we can say that Aswatthaman was a great warrior, but with uncontrollable anger, which led to his downfall.
As far as I understood, there was no symbolism in Aswatthaman's story, except lessons to be learnt.
Drona and Aswatthaman are examples of how a brAhmana, possessing skills of high degree, should not be.
