Richard Martin is correct when he states that "bending a reed" is merely a term to describe a result, rather than a physical action. A bend refers to the pitch, not the reed itself.
BENDING NOTES Certain notes are missing on the standard major-tuned diatonic harp. However, most of those notes can be found by what is referred to as "bending" the reeds, a technique commonly used in playing blues and country. Bending is used mostly on draw notes (except holes 8, 9 & 10, which are blow bends), and results in flatting the note anywhere from 1/2 step (one semitone) to 1-1/2 steps (three semitones). To bend a note, start drawing gently on hole #3 until you get a clear, clean tone. Then drop your jaw slightly, and draw your tongue back toward your throat. (Saying the syllable "tah" will help with draw bends.) This will cause the note to flat, or bend. With practice, you will be able to control the amount of bend, and also you will be able to maintain the pitch. Eventually it will be possible to play the exact bent note directly, without having to bend down to it. Care must be given to not draw too hard, as the reed will not respond properly, and may even fatigue or fracture. (Bending notes fatigues reeds faster than playing straight harp.)
MORE ON BENDING Some notes will bend more than others, and some will bend only slightly, depending on their relationship within the scale. Bends are based on 1/2 step (one semitone) graduations, and are governed by the pitch of the other reed within the same chamber. The pitch of the higher reed can be bent down to one semitone above the pitch of the lower reed. Note that there is a whole tone between C (blow) & D (draw) on hole #1 of a C major harmonica. This means that by drawing on hole #1, it is possible to bend D down 1/2 step (one semitone) to Db. On hole #2, there are three semitones between E (blow) and G (draw), so it is possible to bend the G draw down two semitones to F# and F. Hole #3 has four semitones from G (blow) to B (draw), allowing bends of three semitones from B to Bb, to A, to Ab. Holes #4 & #6 have two semitones each from blow to draw, allowing bends of one semitone each. Holes #8 thru #10 do not have draw bends, but rather have blow bends. Blow bends are done with the jaw slightly dropped, but with the tongue forward, just behind the teeth. (Saying the syllable "tee" will help with blow bends.) #8 will bend one semitone (from E to Eb), #9 will bend one semitone (from G to Gb), and #10 will bend two semitones (from C to B to Bb).In order for a draw reed to bend, the blow reed in the same hole must be lower in pitch. And for a blow reed to bend, the draw reed in the same hole must be lower in pitch. The reason some notes will bend more than others, or only very slightly, is that each reed is dependant on, and interacts with, the other reed in the same hole. The greater the distance in pitch from one reed to the other in a particular hole, the more bend is possible, as in hole #3. Since there is only one semitone difference between reeds in both holes #5 & #7, only a slight bend is possible - less than one semitone each.
Though the above bending documentation is for a C Major harmonica, the patterns are identical for all Major tuned harmonicas.
OVERBLOWS AND OVERDRAWS A relatively new method of obtaining some of the "missing notes" has been established and is being used by a growing number of advanced players. It has been given the descriptive name of "Overblow" and "Overdraw". The effect is that a given note is raised one semitone above the other note within the same hole, using this method. Example: on a C Major harmonica, hole #4 blow, C, can be overblown to D# (one semitone above the draw reed), and hole #7 draw, B, can be overdrawn to C# (one semitone above the blow reed). In order to overblow, the draw reed must be higher in pitch than the blow reed in the same hole. And to overdraw, the blow reed must be higher in pitch than the draw reed in the same hole. Holes 4, 5 & 6 are easiest to overblow, but theoretically overblows can be done with holes 1 thru 6, and overdraws can be done with holes 7 thru 10. Admittedly a difficult effect to achieve, it is best reserved for the very advanced player. Overblow and overdraw notes are usually used only as passing notes, and are extremely difficult to sustain. They also usually lack a "pretty" tone, and sometimes are more of a squeal or squeak by their very nature.