Advantages:

• The are cheap- about $1 at Radio Shack and less from places like All Electronics.  On the low end, only a $10 or less digital volt-ohm meter and a look-up table or graph is required.
• A bit better than 1degF accuracy when I calibrated them against a 1/2 degF or better thermometer.  Better yet, their resistance to temperature (R-T) response from thermistor to thermistor is dead-on so they are interchangeable.
• There are small- a bit smaller than the head on a kitchen match.  This results in them being very easy to mount in a probe and allow them to respond very rapidly to temp. changes.
• A probe can be any length you want- a long one is great for mash probes for assessing temperature gradients  in a mash tun while a short one is ideal for a fixed mounting in a pipe run or to a mash tun or boiler so you're less likely to whack it while stirring.  (A distinct problem with longer stemmed thermometers.
• They are fairly rugged.  The only way I've managed to damage one is by dropping a completed temperature probe so the exposed thermistor at its business end hit on the concrete floor.
• They are immune to electrical noise so you can and can use about any kind of wiring with them.

• Their resistance vs. temperature response is not linear.  This is a very big complication if you want to build a direct reading digital thermometer.  For this sort of application, a diode or a sensor like the LM23 is a better choice.  Linear thermistors are made, but they have temperature range limits and are more expensive.  Info on designing and building you own is in the Links section below.
• For use in liquids, they have to be mounted and sealed in a probe.
• A "maybe"- I don't know what the coating or paint is that covers the bead of the thermistor.  I don't worry about it- very little of it is exposed to wort and I've not noticed any degradation of the coating in several years of use.

A drawing:

I mount them in 1/4" copper or stainless steel tubing so that a lot of the business end of the thermistor is exposed.  This makes them very responsive to changes in temperature.

Here are some very detailed Instructions.

I'll avoid insulting your intelligence by detailing how to connect the probe to a volt-ohm meter (VOM).  BTW, a digital VOM is much better and an analog one for this application.

Calibration/Checking
Since the thermistors I've gotten from Radio Shack have been very uniform (within < 1degF), I treat them as interchangeable and hence there's no need to calibrate each probe.  What I did initially was to put the probe and an accurate darkroom thermometer in an big insulated glass which also had a slowish stirrer in it.  I stated with hot water- around 150 degF and periodically recorded the temp. and resistance as the temperature of the water dropped   The R-T values were essentially the same as those on the back of the Radio Shack pack.  I also dunked the probe in boiling water and in an ice bath and, again, got the resistances on the back of the pack.

Using a Probe
After connecting a probe to a digital VOM set to measure kOhms, it'll display the resistance of the thermistor.  The back of the RadioShack package their thermistors come in have a table of  resistance vs. temperature (R-T) values but it's only in 5 degC intervals.  Not good enough for most homebrewing applications, so make one of the following:

A Table of R-T values
One approach is a table like that is the tab delimited text file Table of resistance vs. temperature  (Hint: save link as a text file on your disk).  It  contains the resistance values for the range 23 to 212 degF, in 1 degF intervals.    Edit/pretty it up  it to your liking with your word processor and print it out or stuff it into a spreadsheet and play around with it.  The NTC app. notes in the Links section below has info on R-T equations which are perhaps more precise than the one I used (a fifth order polynomial).

Graphs of R-T values
1.   Here's a graph that goes from 70-212 degF (on the horz. axis).  (Sorry about the gray background and your waste of toner should you print it, but, I got tired and hence very pissed trying to find where ^&$#@$@ Bill Gate$ hid the background color setting for it in Excel.)

2.   OK, here's a better graph (via a very old version Lotus 123!)

A slick way to mount a proble in a run of 1/2" copper tubing is to hack-saw and grind down one end of 1/4" brass compression union coupling to leave a short nub and shoulder on one end and a compression end on the other.  Drill a hole in the 1/2" tubing to accept the nub and solder the hacked fitting to it.  Plenty strong and cheap.
 

Going Further

Learn more about thermistors- Links
The  Thermometrics web site  has much useful info.  Recommended stuff:

• What is a Thermistor
• NTC Application Notes  (786k PDF file- has info on how to linearize R-T response over a range of temps. and more accurate equations than the one I use)

Sensor Application Notes

Make a simple controller
As much as I like 'puters and Stamps, I'm still an engineer so I try to chose the most appropriate tool for the job at hand.  Since maintaining a beer fridge at a constant temp. is a very easy task, I built a simple temperature controller circuit.