They used a variety of systems usually based on public figures who were in office at the time. Athens, for example, was officially governed by a group of nine archons, who were selected by lot from the richest citizens and held various public offices for the duration of one year. One of these archons was the Eponymous Archon (the title literally means "name-giving leader"), whose name was used to indicate the year in which he served. Greek historical texts will often refer to who was archon at Athens in order to indicate when something happened; inscriptions often mention something along the lines of "...when X was archon" to show when they were set up.

However, this measurement of years would only really mean something to those who knew who was archon at Athens, and other city-states had different ways of counting years. To demonstrate just how varied these systems were before the general acceptance of the BC/AD (or BCE/CE) count, here's how Thucydides dated the start of the Peloponnesian War:

The thirty years' truce which was entered into after the conquest of Euboia lasted fourteen years. In the fifteenth year of the truce, in the forty-eighth year of the priestess-ship of Chrysis at Argos, in the Ephorate of Ainesias at Sparta, in the last month but two of the Archonship of Pythodoros at Athens, and six months after the battle of Potidaea, just at the beginning of spring, a Theban force a little over three hundred strong, under the command of their Boiotarchs, Pythangelos, son of Phyleides, and Diemporos, son of Onetorides, about the first watch of the night, made an armed entry into Plataia, a town of Boiotia in alliance with Athens.

-- Thuc. 2.2.1

There are a few other occasions in his work where Thucydides uses the dating systems of Athens, Sparta and Argos, as well as an independent marker (the beginning of a truce, the date of an earlier battle, etc). We should assume that every single city-state had its own way of keeping track of years in similar ways, but that they were aware of the equivalent year in the dating system of a larger state relevant to them. In addition to this, there was the cycle of Olympic Games, which then as now were held every four years; the Greeks counted them back to 776 BC, and the number of an Olympiad was a solid indication of date.

So, regarding Aristotle, if we flip through Diodoros of Sicily to the year modern editors have identified as 384 BC, we find this:

At the conclusion of the year, in Athens Diotrephes was archon and in Rome the consuls elected were Lucius Valerius and Aulus Mallius, and the Eleians celebrated the Ninety-ninth Olympiad, that in which Dikon of Syracuse won the stadion [foot-race].

-- Diod. 15.14.1

Different places used different systems. So in Athens, each year was named after a magistrate (the eponymous archon). To refer to a given year you'd say "in the archonship of X." In Republican Rome the same thing was done, only with the names of the consuls. Places with kings often used regnal dating, so "in the tenth year of king Y." Sometimes a major event was used as the foundation of a dating system. We know of cities in Macedonia, Syria, and Spain, for example, whose calendars used their conquest by Rome as a watershed moment like our BC/AD system uses the birth of Christ.

This all works fine if you're using a calendar for local administrative purposes, or only really care about local history. If you want to write or think about history on a broader scale, however, it gets really messy, really quick. The Peloponnesian War, for example, involved practically all of Greece, so to date it Thucydides had to cite multiple dating systems:

"[The war began] in the forty-eighth year of the priestess-ship of Chrysis at Argos, in the Ephorate of Aenesias at Sparta, in the last month but two of the Archonship of Pythodorus at Athens, and six months after the battle of Potidaea, just at the beginning of spring." (Thuc. 2.2).

Rather than continue to awkwardly cite dates like this, Thucydides opted for a relative chronology based on seasons in the war (so, "in the spring of the third year of the war"). This works for continuous narrative on a relatively brief time-frame, but for larger time-frames becomes too messy. It's also very hard to line up the dates between different systems reliably. Diodorus Siculus, for example, often made dating errors when he wrote a history trying to synchronize Athenian archonships with Roman consulships.

Ultimately it was decided to use the dates of Pan-Hellenic games, and above all the date of the Olympics, for pan-Mediterranean dating. The first historian to use Olympic dating (as it is called) was a third century BC Sicilian named Timaeus. But even after the rise of Olympic dating, other methods continued to persevere, especially at the local level. Our modern system (called incarnational dating) was developed in the Early Middle Ages.