10BCE happens to be very convenient for your question, at least in the context of Rome. The Julian calendar, appropriately named after Julius Caesar, came into effect only 36 years before your chosen date. It is identical to our modern, Gregorian caledar, save the removal of some leap years; western astronomy wasn't quite as advanced as it became later, and it turned out that a solar year wasn't quite precisely 365.25 days.

You might be unfamiliar with one particular month name, however; Quintilis was already renamed after Caesar's death in his honour and became Iulius - July in the English tongue, but the senate would not rename Sextilis in the honour of Emperor Augustus until two years after your chosen reference point.

Now, the crux of the question, the year. Before the advent of Christianity, and even well after its rise, you wouldn't really think of a year as a number. Counting years from the founding of Rome did occassionally show up in a contemporary context, but was hardly in every day use. If you wanted to refer to a year in that particular period, you would refer to it by the name of the two consuls who stepped into office that year. Indeed, Rome under Augustus was still, at least nominally, a republic, although a discussion on how true that was is outside of my scope of expertise. Wikipedia provides a handy list of consulateship, and if we check there, 10BC was the year of consuls Africanus Fabius Maximus and Iulius Antonius.

While I haven't been able to find a Latin source for 10BCE specifically, I have a passage from "The Deeds of the Divine Augustus." It is monumental inscription written from the perspective of Augustus himself, and it should hopefully illustrate how this would look in practice:

Annos undeviginti natus exercitum privato consilio et privata impensa comparavi, per quem rem publicam a dominatione factionis oppressam in libertatem vindicavi. [Ob quae] senatus decretis honorificis in ordinem suum me adlegit, C. Pansa et A. Hirtio consulibus, consularem locum sententiae dicendae tribuens, et imperium mihi dedit.

or, in a more familar tongue, translation courtesy of Thomas Bushnell:

In my nineteenth year, on my own initiative and at my own expense, I raised an army with which I set free the state, which was oppressed by the domination of a faction. For that reason, the senate enrolled me in its order by laudatory resolutions, when Gaius Pansa and Aulus Hirtius were consuls, assigning me the place of a consul in the giving of opinions, and gave me the imperium.

Looking back at our list, Gaius Pansa and Aulus Hirtius were consuls in what we now know as 43BCE, and this would have been clear to a contemporary reading the monument.

While this example is specific to Rome, the notion of naming years after rulers was hardly unique to them. A notion of a regnal calendar would pop up time and time again: the practice of counting the reigning years of the local form of big, important rules. For example, such a system would pop up in Japan several centuries later, and curiously, still persists to this day in many contexts: 2022 is Reiwa 4, corresponding to the fourth year of emperor Naruhito's reign.

For another example, the earliest Japanese calendar was an import from China. The specifics of the calendar are quite different to what we are used to nowadays, being a combination of a lunar and solar calendar, but years were still years; they are all ultimately based on orbital dynamics.

There, too, a regnal calendar was often used. 10BCE would be the 3rd year of the Yuanyan era... but Yuanyan was not the emperor, but an era name the ruler chose. The emperor at the time was Cheng of the Han dynasty, but this name was only given to him posthumously. His given name was Ao, but speaking the name of the living emperor was considered a massive taboo in China of the time. He would have simply been "the Emperor."

It's not all rulers, mind you. The Mesoamerican civilisations had a history of intricate, complex ritual calendars, including the long count calendar you may recall from the hysteria of 2022, but I am not well read up on that particular region, so I can't say too much on them. I'm afraid you will have to research those yourself, should interest strike you.