(The SH in this name is pronounced like one between S in 'sup' and SH in 'shut', like in SHIVA. The DH is pronounced like a hardened 'th' in 'that'.)
Ashva means horse. Ashvamedham is a sacrifice to be performed only by kshathriyas. The name implies that a horse is prominent in this sacrifice. Also, there are sacrifices named Naramedham and Sarvamedham as referred by vyaasa to Yudhishthira in the beginning of Ashvamedha parva.
Like other Vedic sacrifices, it has its own elaborate procedure lasting for many days. The prime features involve a horse purified with sacrificial processes. The horse is let free to wander around all the kingdoms, as a symbol of the performer's supremacy. An able hero follows it all the while. Those who do not accept the supremacy would seize the horse which is a symbolic act of challenge. The hero in guard of the sacrifice is to defeat the challenger. Then only should the performer be accepted as supreme leader of all the other kings. After the horse returns safe to its sender, some more rituals are performed, in which the horse is killed and its various parts are offered in the fire of sacrificial altar. The performing king breaths in the smoke and scent from that fire consuming those parts of the horse. Also, during the procedures, the king's principal wife should attend on the horse in person. The process involves covering of the queen and dead horse with a large cloth. This stipulation made some foolish critics to imagine that the dead horse and the queen would have coition ! Some ambiguous verses in Braahmanams and Vedas led even the learners to comment upon them like that.
Many ancient kings ( even Vasudeva, who is not a king ) are mentioned as performers of this celebrated sacrifice. In this epic, Yudhishthira performs this sacrifice to cleanse himself from the sins of killing his own kins in the war.
The scriptures proscribe certain things ( like eating meat, Niyoga dharma etc) in this Kali yuga. Ashvamedha sacrifice was one among them. But, we come across names of certain kings in our history who were said to have performed this sacrifice.
"After praying to Naaraayana, Nara, Sarasvathi and Vyaasa, then one should read the Jayam."
DEDICATED TO MY DEAR NIECE VYSHNAVI KRISHNA