There are, generally speaking, three schools of thought on what the seasons are. These are for the Northern Hemisphere; reverse them for the Southern Hemisphere.
-Meteorological seasons, where winter is the entirety of December - February, spring is March - May, summer is June - August, and fall is September - November
-Astronomical seasons, where winter is about December 21 - March 20, spring is about March 20 - June 21, summer is about June 21 - September 23, and fall is about September 23 - December 21
-Solar seasons, where winter is about November 7 - February 6, spring is about February 6 - May 6, summer is about May 6 - August 6, and fall is about August 6 - November 7, using the solstices as midpoints of summer and winter.
Seasons are generally thought of in non-tropical climates as "summer = hot", "winter = cold", and in between is spring and fall. For most climates, the meteorological system is the best approximation of this (except for many oceanic climates, where seasonal lag is strong enough that the astronomical seasons are often the best approximation). Solar seasons are ridiculous (where in the world is there a Northern Hemisphere climate where November 10 is colder than February 10?)
But these three classifications are just that: approximations. Seasons don't start precisely at a solstice or a month boundary. In most of the world, they're defined by temperature. The precise temperatures that define "summer" and "winter" are going to vary wildly from climate to climate, so it would make sense that if the average temperature is within a certain percentage of the temperature in the warmest month (the height of summer) that it would be called "summer", and vice versa for winter. Weatherspark uses 20 percent, which seems like a reasonable number.
For my climate (Bowling Green, KY), the coldest month (January) averages at 34 F / 1 C (high 43 F / 6 C, low 25 F / -4 C) per Intellicast. The warmest month (July) is 79 F / 26 C (high 89 F / 32 C, low 68 F / 20 C). That gives us a 45 F / 25 C range.
(Intellicast and Weatherspark are excellent sources, because they give an average high and low for every day of the year).
20 percent of that is 9 F / 5 C, which means that "winter" is when average temperatures are below 43 F / 6 C, and "summer" is when average temperatures are above 70 F / 21 C. Going through Intellicast, we find that the average temperature crosses that 43 mark on March 2 and November 30.
The 70 mark is surpassed on May 30 and dropped below on September 15. This means that for Bowling Green, KY:
Winter: November 30 - March 1
Spring: March 2 - May 29
Summer: May 30 - September 14
Fall: September 15 - November 29
which is pretty close to the meteorological reckoning, with the exception of late summer.
San Francisco is well-known for its summers being delayed. With a January average at 52 F / 11 C and a September (!) average of 64 F / 18 C, its 12 F / 7 C spread is barely enough to say it has "seasons", but anything within 2 1/2 degrees of those averages is our threshold (below 54.5 F for winter, above 61.5 F for summer)
Winter: December 2 - February 13
Spring: February 14 - July 24
Summer: July 25 - October 27
Fall: October 28 - December 1
Fairbanks, Alaska is on the other end, with seasons that peak early. January, the coldest month, averages -6 F / -21 C and July, the warmest month, is 62 F / 17 C. That's a 68 F / 38 C spread, which means that anything within 14 F / 9 C of those peaks is winter or summer (i.e. below 8 F / -14 C for winter and above 48 F / 9 C for summer).
Winter: November 6 - March 6
Spring: March 7 - May 14
Summer: May 15 - September 9
Fall: September 10 - November 5
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