How can you measure the strength of someone biting, say an apple. Do
dentists do this? Is there any special equipment they use? Can they
put a device in your mouth to see how well you are chewing?

speechmark-ga

This is quite an interesting question.

To start off, some background on bite-force with relation to age:
"In a comprehensive study of oral function in humans the masticatory
performance was determined for a group of 814 subjects between 25 and
75 years of age. The investigation was limited to persons with natural
dentition or with missing dentition replaced by a fixed  prosthesis.
According to this study, masticatory performance does not alter
significantly with age in persons who have a complete or almost
complete  dentition. However, in these persons, there is a significant
increase with age in the number of strokes used to prepare the test
food for swallowing. Although maximum bite force appeared to be
significantly larger in a group of young dentate subjects as compared
to old dentate subjects, no significant differences in masticatory 19
efficiency were observed . Apparently, the lower maximum bite force of
the older subjects was still high enough for chewing the test food."
Also, North American Eskimos can exert up to 360 lb of pressure; the
average person can exert up to 125-150 lb of pressure; denture wearers
exert only 15-17 lb of pressure.

Bite force is important to dental engineers as they must ensure that
the dental replacement they are designing can withstand the forces
which the mouth places on them, around 20-500 N, depending on the
direction of the force.

One method of accomplishing this this task, though slightly
complicated, would be to measure the size and orientation of the
masseter muscle to determine what force it COULD exert on foods of
various sizes.  Using and MRI, the size and volume of the masseter
muscle can be determined, the chief muscle used in chewing and biting.
However, this is expensive an probably not what you are looking for.
(http://nr.stic.gov.tw/ejournal/ProceedingB/v25n1/45-49.pdf)

Another method that is utilized to measure bite force is find a
bitable material, scientifically determine the force require to
penetrate the said material.  The values of force required to
penetrate certain foods are available online and the following is a
sample of such a table
Boiled Meat - 80N
Carrot - 118N
Rye Bread - 167N
In this way, the bite force of the individual can be easily measured
by slowly incrementing the toughness of the food.
(http://www.fop.unicamp.br/brjorals/BJOS-v1-n1-p7-18.pdf)

The way that anthropologists measure bite force is by reconstructing
the muscle on old bones found of Neandertal and used a digitizer, a
machine that gives the three-dimensional coordinates for a particular
point, to enter into a computer the size and attachment sites of each
muscle. By applying standard engineering formulas to the data, they
could then calculate how much force each muscle produced, how much
force was placed on the jaw joint, and how much total force went into
the bite.
http://scicom.ucsc.edu/SciNotes/9801/stick/3.html

To answer you question about dentists, yes, they do measure bite force
of a patient, but their normally goal is slightly different from your
question.  They are less concerned about the actual strength of bit,
and more concerned about the various forces in relationto the teeth. 
In other words, they want to know how the teeth come together, and the
force on the individual teeth.  The usual equipment used by dentists
and orthodontists to do this is usually a wax-mold or pressure
indicator pastes, which change color to show the pressure exerted on
it.  Thus, one could easily take such a paste, place it between two
flat surfaces so the force is uniformly distributed and have somebody
bite down on it to see the bite force.
Dentists are concerned about actual bite force after surgery on the
mouth or if a new set of dentures is installed to check if th
procedure was done properly.  Low bite force would indicate a
misalignment in the teeth, or even worse, damaged nerves inside the
mouth, making controlling the masseter muscle difficult. 
(http://www.dent-links.de/science/2000/2000_10759115.htm) Generally, a
sensitive pressure indicator sheet is used to check the bite force in
this case.  This pressure-sensitive device can be used as a simple
indicator of postoperative recovery and occlusal improvement in
patients following orthognathic surgery.

So in summary, dentist and orthodontists DO utilize tools to measure
bite force of patients. In general, they use either pressure-sensitive
sheets which change color depending on pressure applied, or they use
wax, and from the indentations, the bite force can be calculated.

However, these are not the only tools available for measuring bite
fore.  Others include commercially available products such as Tekscan
(http://www.tekscan.com/dental.html) which uses a handheld device
which attaches to the computer for analysis of many of the forces (not
just bite force) in the mouth.  A description of another bite force
meter is available here
(http://www.dt.mahidol.ac.th/Journal/DENT19(1-3)-3.pdf)

So as you can see, there are many ways to measure bite force!

Hope that helps, if you have any questions, just ask for clarification
and I'll be glad to help you.

-Tox-ga

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Thank you! So fast. I'm thinking of designing a simple tool along the
lines of one of those you mentioned so you have saved me hours of
searching!