A quarter hour is a unit of time equal to 15 minutes. The terminology comes from there being 60 minutes in 1 hour. Thus, a quarter, or ¼ (or 0.25) of 60 minutes is 60/4 = 15 minutes.
The term quarter hour is most commonly used in the context of telling time. For example, "quarter past" indicates that it is 15 minutes past the current hour, while "quarter till" means that it is 15 minutes before the next hour. The term "quarter to" is also used in place of "quarter till."
If the time is 10:15, this can be said as "quarter past ten." 10:45 on the other hand would be said as "quarter to/till 11." Although you could theoretically indicate 10:30 as "two quarters to eleven" or "two quarters past ten," these expressions of time are not used in everyday language. Instead, "half past 10" is likely the most common usage next to saying "ten-thirty."
Analog vs. digital clocks
Reading time off of a digital clock is relatively simple compared to reading time off of an analog clock. However, it is easier to indicate specific periods of time (such as a quarter-hour) on an analog clock since analog clocks are most typically designed as circles, and can be broken up into slices/regions that are likely familiar to those who have studied topics such as fractions, ratios, and proportions. Below is a representation of a quarter hour passing on an analog clock, from 8:00 to 8:15.
The time above can either be read as "eight fifteen" or "quarter past eight." Note that the use of "quarter" as a modifier of a measurement of time is largely only used when telling time using minutes. For example, while it would technically be correct to say that a period of 15 seconds passing on a clock is a "quarter minute," this is not commonly used. "Quarter day" is also not used to describe a period of 6 hours.
Did you know?
Notice the "quart" in "quarter." There are four quarts in a gallon, four quarters in a dollar, four quarter-hours in an hour, and four quarters in a football game. Quart comes from a Latin word meaning four or fourth.