The summer solstice takes place on June 21 (or 22nd):
That makes the day the longest of the year.
Now: Which day is the second longest of the year?
Is it the day before or the day following the 
Summer Solstice? Explain. 
(The answer is NOT that their both the same.)

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Request for Question Clarification by thx1138-ga on 28 Jun 2002 15:17 PDT

Are you in the Northern or Southern hemisphere ?

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Clarification of Question by streetlight-ga on 29 Jun 2002 09:39 PDT

Let's use the Northern Hemi to clarify the question. (As the question
deals with the longest day
and not the shortest.)  Yes, I agree its not a simple problem. 

 Whatever, there cannot be two successive days of exactly the same
time for rise and set times.

That's not as simple a question as it sounds!  As the Request for
Clarification suggests, in the Northern Hemisphere is is the Summer
Solstice and longest day of the year; But in the Southern Hemisphere,
the June Solstice is the shortest day of the year, and falls in the
dead of winter.  Of course, as one moved around and between the two
hemispheres, some measure of in-between would be true.  To quote
Window's to the Universe's 'advanced' page on the Summer Solstice:

"How high the Sun gets in your sky, and how long it is above the
horizon during the day, depend not only on the season, but also on
your latitude."

( http://www.windows.ucar.edu/tour/link=/the_universe/uts/summer.html&edu=high
)

That page provides a helpful illustration of the various factors
affecting the solstice.  I hope it helps you or another researcher
answer your question!

Observer's Location  is New York ( 40.7140°N, 74.0060°W)
“.On June 21, 2002 at 8:24 am CDT is the moment of the summer
solstice”
http://216.239.35.100/search?q=cache:kgR_sEwCrYcC:www.scienceplace.org/PL/PLAstronomicalInfo.html++%22moment+of+the+summer+solstice+%22&hl=pt&ie=UTF-8

8:24am CDT = 9:24  EST


By calculating the times of sunrise and sun set using the astronomical
calendar here:
http://www.heavens-above.com/sun.asp?lat=40.714&lng=-74.006&alt=2&loc=New+York&TZ=EST&Date=37426.9368402778
we see that :

Date/June          Sunrise                  Sunset
19                 5:24                     20:30
20                 5:24                     20:30
21                 5:24                     20:30
22                 5:25                     20:30
23                 5:25                     20:31
24                 5:25                     20:31

The 20th of june is the second longest day (by one minute!) because
the moment of the solstice was at 9:24 EST after which time the sun’s
zenith crosses the tropic of  cancer and the days start getting
shorter (in the Northern Hemisphere) Therefore the second longest day
was the 20th of June, and not the 22nd.

That's my reckoing Anyway!

My guess is that the day could vary from year to year depending on the
exact time of the solstice.  Just a guess.

The simple answer is the day that is closest to (but not on the same
day as) the precise time of the summer solstice.

The example posted by thx1138 would mean for NY, June 20th would be
the second longest, since 8:24am EST is closer to the 20th than 22nd. 
Note the use of EST instead of EDT.  You can use Daylight Savings Time
if the dividing line is at 1 pm.

Think of the length of daylight though the year as a slow (sine?) wave
graph.  Solstice, meaning sun standing still, are the top and the
bottom of the wave.
The actual day for summer is a 24 hour period spanning that peak
(depending what longitude you are at).  The next longest day would be
that day on either side that is closest to that peak.  Of course if
you are above the polar circle, the sun doesn't rise or set on the
sumner solstice.  You may have many 24-hr periods which the sun
doesn't set.

http://www.religioustolerance.org/summer_solstice.htm
http://www.jgiesen.de/astro/solarday.htm
http://aa.usno.navy.mil/data/docs/RS_OneYear.html

The longest day of the year does not mean the earliest sunrise or the
latest sunset.  There is more going on.

You asked (with my numbers added): 

1) Which day is the second longest of the year? 
2) Is it the day before or the day following the  
Summer Solstice?

To answer your *2* questions:
1) The day on which the Winter Solstice occurs in the southern
hemisphere.
2) No

Or, assuming you didn't mean to ask a trick question...
The answer provided by thx and ulu appear to be correct.
If you just happened to be located at the exact latitude line on the
Earth that perfectly faces the sun at the moment of solstice then both
the day before and the day after the solstice would be exactly the
same length for you.  This is due to the symetry created by directly
facing the sun at that point in time.
If, as is more likely to be the case, you're not standing on that
latitude line then the day before and the day after will have
different lengths.
In order to determine which day is longer you will need to know if you
are standing east or west from this magical line (or a third case
involves standing exactly 180 degress away, but in this case two
consecutive days would share the longest day award).  If you remember
that the sun "rises in the east" then you know that the day has
already passed its peak for those standing to the east of the line. 
These "easterners" will think the day before was the longest, while
the "westerners" will disagree.