Inside your mouth are taste receptors. They pick up on the senses of salty, bitter, sweet, sour, and umami (which seems to be activated by monosodium glutamate (MSG)). These receptors live inside tastebuds, which in turn live inside the tiny bumps you feel on your tongue (called papillae). Different receptors are better at picking up these different tastes, and send these signals via neurons to the brain, giving us the sensation of taste. In the papillae, wrapped around the tastebuds, there are also a set of neurons that send information about temperature, and pain. These are the receptors we are going to concentrate on.

Now say you take a big bite into a jalepeno pepper. Why does it feel hot ? The reason is that hot peppers contain a molecule called capsaicin. This molecule basically excites a small set of the pain sensors which also happen to respond to hot temperatures. So basically your brain gets the signal that there is something hot/painful there, even though there really isn't. Other types of spices such as say menthol found in peppermint, for example trigger a different set of receptors, ones that send signals for pain and cold.

One way to think about it is that the temperature receptors are like a lock, and that hot stuff is the key. When the key meets the lock, the neuron carries a signal to the brain. Now say we have don't have the key but we have something that is shaped pretty close to it. We will still be able to open the lock, thus tricking the lock into thinking we have the key. This is different from say, smashing the lock open with a hammer. In other words, though there is a chemical setting off these receptors, it isn't actually a chemical burn or actual damage that causes the burning/painful sensation.

One final note, which is partially from personal experience : Capsaicin triggers a type of receptor which can be found anywhere on the body. If you've ever cut really hot peppers, you might notice your hands starting to tingle/burn slightly. This is the capsaicin activating the same type of receptor, only on your hands. It is also a good reason to wear gloves when handling hot peppers. In addition, this is why it is a bad idea to rub your eyes after handling peppers (ouch !).
http://www.madsci.org/posts/archives/2001-06/992019854.An.r.html

Spicy foods contain a chemical called capsaicin; the spicier the food, the more capsaicin it contains. This chemical interacts with a protein called VR1, or vanilloid receptor 1 (vanilloid = class of chemical compounds that includes capsaicin). Activation of this receptor on neurons (nerve cells) causes depolarization (loss of net ionic charge across the cell membrane) and entry of calcium ions, signalling that a spicy substance has been encountered. VR1 is an incredibly cool protein for a couple of reasons, outlined below.

First of all, VR1 is responsive to other stimuli, as well. Capsaicin activates it strongly, but VR1 also responds to heat. The reason why spicy foods and hot foods both feel "hot" is because both stimuli are detected by the same molecule. VR1 also responds to a combination of heat + spice, which is why warm Mexican or Thai food feels even "hotter" than cold spicy food. VR1 is also sensitive to acid pH. Acid won't open the channel on its own, but it will lower the threshold for responding to heat or spice. The reason why acid spilled on the skin feels like it "burns" is because the acid enables VR1 to respond to the skin's own temperature of 37 degrees C. Incidentally, some animals don't sense spicy food (like birds, which consume peppers and then disperse their seeds miles away, in new habitats). VR1 researchers speculate that their VR1 genes might be different.

Secondly, VR1 is present in all the right places to be responsive to spice and heat: the tips of neurons that detect painful stimuli to the tongue, mouth, skin, and so forth. VR1 is present in other neurons, as well, which suggests that it may have roles in addition to its heat/spice/ pH-detecting roles.

Third, VR1 activation allows calcium and other ions to enter neurons. Calcium is an important signaling molecule; it can allow cells to release neurotransmitters, or to make changes in the genes they express. The signaling events downstream of VR1 activation are being explored, and they will probably be interesting!

Finally, VR1 is related to a growing family of human, rodent, fly, and worm channels, which appear to be involved in various sensory functions (vision and smell, for starters). Understanding the basis of sensory function in any of these model systems will probably shed light on how VR1- type channels work, in general.

Here are some good references on VR1 and related proteins:

From http://www.nature.com , the original identification of VR1:

Nature 389, 816 - 824 (1997) © Macmillan Publishers Ltd.
The capsaicin receptor: a heat-activated ion channel in the pain pathway
MICHAEL J. CATERINA, MARK A. SCHUMACHER, MAKOTO TOMINAGA, TOBIAS A. ROSEN,
JON D. LEVINE & DAVID JULIUS

Capsaicin, the main pungent ingredient in 'hot' chilli peppers, elicits a sensation of burning pain by selectively activating sensory neurons that convey information about noxious stimuli to the central nervous system. We have used an expression cloning strategy based on calcium influx to isolate a functional cDNA encoding a capsaicin receptor from sensory neurons. This receptor is a non-selective cation channel that is structurally related to members of the TRP family of ion channels. The cloned capsaicin receptor is also activated by increases in temperature in the noxious range, suggesting that it functions as a transducer of painful thermal stimuli in vivo.

A follow-up paper at http://www.neuron.org :

Neuron, Vol. 21, 531–543, September, 1998, Copyright © 1998 by Cell Press
The Cloned Capsaicin Receptor Integrates Multiple Pain-Producing Stimuli

Makoto Tominaga1, Michael J. Caterina1, Annika B. Malmberg2, Tobias A.
Rosen1, Heather Gilbert2, Kate Skinner2, Brigitte E. Raumann1, Allan I.
Basbaum2, and David Julius
Department of Cellular and Molecular Pharmacology, W. M. Keck Foundation
Center for Integrative, Neuroscience, University of California, San Francisco,
California
Departments of Anatomy and Physiology, W. M. Keck Foundation Center for
Integrative, Neuroscience, University of California, San Francisco,
California

Capsaicin, the main pungent ingredient in ''hot'' chili peppers, elicits burning pain by activating specific (vanilloid) receptors on sensory nerve endings. The cloned vanilloid receptor (VR1) is a cation channel that is also activated by noxious heat. Here, analysis of heat-evoked single channel currents in excised membrane patches suggests that heat gates VR1 directly. We also show that protons decrease the temperature threshold for VR1 activation such that even moderately acidic conditions (pH 5.9) activate VR1 at room temperature. VR1 can therefore be viewed as a molecular integrator of chemical and physical stimuli that elicit pain. Immunocytochemical analysis indicates that the receptor is located in a neurochemically heterogeneous population of small diameter primary afferent fibers. A role for VR1 in injury-induced hypersensitivity at the level of the sensory neuron is presented.

In addition, you might check out the home page for the lab which identified VR1 and related channels.
http://www.madsci.org/posts/archives/1999-12/946489338.Ch.r.html