Seasons

The primary cause of seasons is the Earth's obliquity, the 23.5 degree tilt of the Earth's rotation axis relative to the ecliptic plane.

Plane parallel atmosphere approximation

Assume the local surface of the Earth is flat, as is the thickness (H) of the atmosphere above the surface.  The solar intensity at the top of the atmosphere is the amount of sunlight (per unit area) that is incident on the top of the atmosphere (Itop).  But some amount of that sunlight is absorbed in the atmosphere and doesn't make it to the surface; thus, the amount of heat that reaches us depends on what happens to the sunlight as it passes through the atmosphere.  What happens in the atmosphere is that a great amount of sunlight is scattered and absorbed (we call this extinction).
If the Sun is directly overhead, sunlight passes through this atmosphere with a minimum of attenuation.  If the Sun is at an angular distance z from the zenith (straight overhead), then sunlight has to pass through an atmospheric column whose length s is longer than H and is given as s = H/cos(z). The total amount of sunlight that is removed is proportional to s and depends on the amount of extinction k per unit distance along the light path through the atmosphere.  Mathematically, the sunlight that reaches the Earth is found as:

Iground = Itop   exp[-ks] = Itop   exp[-kH/cos(z)] = Itop   (exp[-kH])1/cos(z)

The changing distance from the Sun, caused by the Earth moving from perihelion (closest approach) to aphelion (furthest distance) in it's elliptical orbit around the Sun is not a primary contributor to seasonal change.  The dates of perihelia and aphelia change each year.  Exact dates can be found here.

Arctic circle: 23.5 degrees from the north pole.
Tropic of Cancer: 23.5 degrees north of equator.
Tropic of Capricorn: 23.5 degrees south of equator.
Antarctic circle: 23.5 degrees from south pole.

At northern hemisphere midsummer (the summer solstice, June 21), the sun would be directly overhead (90 degrees from all horizon directions; the zenith) at noon as seen from a latitute of 23.5 degrees north (Tropic of Cancer).  The elevation of the Sun(above the horizon at this date would be 23.5 degrees (24 hours per day) for someone at the north pole.  For someone at northern latitude X, the highest point of the Sun in the sky at midsummer is 90-X+23.5

At southern hemisphere midsummer (the winter solstice, December 21), the sun would be directly overhead (90 degrees from all horizon directions; the zenith) at noon as seen from a latitute of 23.5 degrees south (Tropic of Capricorn).  As seen from the north, all angles change by 47 degrees.  So the highest elevation of the Sun above the southern horizon, in midwinter for someone in at northern latitude X would be 90-X-23.5.

In Nashville (northern latitude 36 degrees), the height of the Sun above the southern horizon at midday on December 21 is 90-36-23.5 = 30.5 degrees; on March 21 (vernal equinox) and September 21 (autumnal equinox) the height above the southern horizon reaches 54 degrees; on June 21 (summer solstice) it reaches 77 degrees.

At the north pole, the elevation of the Sun at midday,  changes from 0 (vernal equinox) to 23.5 (summer solstice) to 0  (autumnal equinox) to -23.5 (winter solstice).

From Nashville, the elevation of the Sun changes from 54 (V.E.) to 77.5 (S.S.) to 54 (A.E.) to 30.5 (A.S.).

At the Tropic of Cancer,  the elevation of the Sun changes from 66.5 (V.E.) to 90 (S.S.) to 66.5 (A.E.) to 43 (A.S.).

At the equator, the elevation of the Sun changes from 90 (V.E.) to 67 (S.S.) to 90 (A.E.) to 67 (A.S.).

Question: How would the seasons be different if the Earth's obliquity were smaller? larger?

Precession

If Earth's rotation axis were fixed in space, the seasons would be fixed in time, relative to an orbital period of the Earth around the Sun.  But because of precession, the seasons move around the Earth's orbit.

Imagine that the rate of precession is one cycle in 10 years.  Then in 5 years, the rotation axis has precessed 180 degrees from its current position.  Thus, if today June 21 is the first day of summer, then in 5 years June 21 would be the first day of winter.  The seasons would migrate around the calendar in 10 years and clearly the length of the seasonal year would be different from an orbital period.

Because the precession rate is so small:
     period = 25,770 years
     rate per year = 360 degrees/25770 years = 0.014 degrees/year = 0.84 arcminutes/year
the seasonal (tropical) year and orbital period are nearly the same.