I know that holidays begin at sundown the day before the date they are on. Do Jewish people begin to celebrate the holiday at sundown or on the date they are on? Take for example, Passover. Are any of the Passover meals eaten at sundown of the day before the holiday starts? I would also like to know about any differences from modern day and in antiquity.
For the entirety of Jewish History, Judaism has kept to a day that begins at sunset on the previous 'secular/non-Jewish date'. Therefore, all meals eaten after sundown would be considered to already be eaten on that day, not on the previous day.
What is very interesting about your question is that the Torah specifically mentions the eating of a Passover meal on the Eve of Passover (according to the traditional interpretation):
In the first month, from the fourteenth day of the month at evening, you shall eat unleavened bread until the twenty-first day of the month at evening.
Exodus 12:15 (Sefaria translation)
An English article that touches on the sources discussed here can be found here.
Holidays don't begin before the date they are on. They are only on the date they are on. That date starts and ends at sundown. Many holiday rituals occur after sundown at the beginning of the holiday.
Modern common calendars by popular convention begin and end dates at midnight, so the holiday's date overlaps partially with two of the modern common calendar's dates. The publisher could mark both dates as overlapping partially, but since the second modern common calendar date overlaps with most of the holiday, they often just mark that. It's imprecise, but usually sufficient for most people's needs.
As others have noted, this has been a very longstanding Jewish rule - holidays start/end at sundown. But it actually gets a little more complicated. A brief overview:
1 - This applies to ALL days, not just special holidays. In particular, that means the days of the week, including Shabbos (Sabbath) start/end at sundown. That means the Friday night special Shabbos dinner (which traditional Jews eat every Friday night, except if Yom Kippur starts on Friday night) is eaten after sundown Friday night and Shabbos continues until sundown on Saturday night.
2 - The time for sundown is "fuzzy". Officially this is called Bein Hashmashos. The ends are "shekia" = "sunset", which generally has a pretty clear astronomical definition (sun below horizon) and "tzeis hacochavim" = "stars come out". Depending on location (ignoring the really wild situation in extreme northern/southern latitudes) and time of year and different traditions/interpretations of the Jewish laws involved, this time is typically between 42 and 72 minutes.
3 - The sundown time is normally treated as an extension in whatever way is strictest. For Shabbos and holidays where work is prohibited, that effectively means a 25-hour "day" instead of just 24 hours.
4 - For normal Shabbos and holidays, one is allowed to start the evening services and the festive meal early (Plag Hamincha). But specifically for Passover the Seder meal is to take place "at night" which is normally understood to mean "after tzeis hacochavim". The result is that Melachah (work) is prohibited from "shkia" but you can't start the Seder until "tzeis". That really isn't a problem as there are prayers and last-minute preparations that can be done "bein hashmashos".
While the previous answers are correct that Jewish holidays begin at night, they do not completely address the question. The question here asks whether all aspects of the holiday (such as meals) begin at night. The truth is that though holidays begin at night, there are often certain aspects that according to Jewish Law can only be done during the day.
- The blowing of the Shofar on Rosh Hashana can only be done during the day even though the holiday starts the night before. (Shulchan Aruch O.C. 588:1)
- The taking of the Lulav on Sukkot can only be done during the day even though the holiday starts the night before. (Shulchan Aruch O.C. 652:1)
- The festive meal eaten on Purim can only be done during the day even though the holiday starts the night before. (Shulchan Aruch O.C. 695:1)