Ancient History dates from the beginning of recorded history under the Sumerians and their Cuneiform script sometime around the 3rd millennium BC until sometime around the 5th century AD depending on what part of the world you're looking at. The end of the ancient period and the beginning of the Medieval period is in Europe clearly associated with the fall of Rome, whichever 5th century date you attribute this phenomenon to.* In China or the Middle East the dates are different, but I'm not sure what events they are generally associated with.

As a side note, Classical history, which is my area, is a relatively short but dominant field of Ancient history focusing on the Greco-Roman world, defined by the beginning of the Homeric era in Greece around the 8th century or so BC until the fall of Rome. Due to the predominance of the Classical period in Ancient history and the cultural and intellectual traditions descended to the West from the Greco-Roman world, when most people say "ancient" or refer to the "ancients" they are referring to the Classical period and its Mediterranean inhabitants.

Essentially, Ancient history refers to a period of time roughly 3500 years long from the beginning of recorded history to the mid/late 5th century AD.

• The traditional date is of course AD 476, when Romulus Augustulus was deposed by Odoacer, but this date is notoriously arbitrary due to the fact that Roman imperial authority in the West had fragmented well before this date. Other dates might include the sacks of Rome in 410 and 455, the appointment of the insidious Ricimer to the post of magister militum in 461, the death of Julius Nepos in 480 or the brutal overthrow of Odoacer by Theodoric in 493.