Ancient history, in European history, is a term referring to the time period between the beginning of writing (and civilizations) and the Middle Ages. It may also be applied to non-European areas, with more unclear ending dates.
"Classical antiquity" (also the "classical era", "classical period", or "classical age") is the period between the 8th century BC (the earliest-recorded epic Greek poetry of Homer) and the 5th or 6th century AD (the fall of the Western Roman Empire), comprising the interlocking civilizations of Ancient Greece and Ancient Rome known as the Greco-Roman world.
"Late antiquity" is a recent term, for a time period between the 3rd and 8th centuries AD, notably used by those disliking the idea of civilizational decline and collapse (especially as stated by Edward Gibbon). Another reason may be that the focus on the (Western) Roman Empire is viewed as Eurocentric and thus not politically correct, with "late antiquity" instead ending with the rise of Islam.