I thought a harmonica is not only played by the mouth, but by the fingering over the holes on the far side (distal to the mouth). Yet, if this is true, how can Billy Joel, for one, play piano and harmonica simulataneously?
Note: I am not asking how he holds it. It is well known he wears some kind of support. The question is if the fingers are needed, too? Or, are there different types of harmonicas?
The fingers aren’t needed for basic harmonica play. They are really used to get the “waver” type effect or for some fancy stuff. BJ does not use a special type of harmonica or anything like that.
The Doctor
I’m not a harmonica player, but I’m sure some will be in here shortly to elaborate. In the meantime . . . AFAIK a harmonica is not “fingered” per se to produce the notes. You blow and draw air throught the holes to produce notes. Any handwork would be to produce vibrato, that distinctive “wah-wah-wah” sound and such, and so forth. So you can play a harmonica reasonably well without using your hands.
On the other hand, I once asked a harmonica player with a thick mustache and beard how he ever managed to use the bracket without doing himself an injury by pulling out facial hairs. He said, “It ain’t easy.” 
There is at least one type of harp, the chromatic harmonica, that does make use of the fingers in order to switch the airflow around which set of reeds you’re playing, but most harmonica players you’re familiar with use the standard diatonic harp, which does not require the use of hands. Hands are used, as Dr Who said, to control the timbre of the instrument (think of how trumpet mutes affect the sound of that instrument) and to create tremolo effects.
It is not at all necessary to use the hands. Haven’t you ever seen Bob Dylan or any of a number of folk or pop guitarists play guitar and harmonica at the same time?
Technically, that’s a tremolo, as you are not altering the frequency of the note, but rather its amplitude (loudness) and tone. Vibrato involves slight changes of frequency, and this is done in the mouth/throat when playing a harmonica. You can apply vibrato and tremolo at the same time.
Yes. Your basic “Bob Dylan” type harmonica only has the notes to play in one key* and you can play it hands free (like his Bobness Neil Young and folkies). There is another type of harp – chromatic, the sort of thing Larry Adler played – which can play in any key (if you like it has all the white and black keys on a piano) but you need to push a shift(?) key with your fingers to get the black notes.
*Blues players use them in the wrong key to get the proper blue notes. That’s how Little Walter can sound that way, and Bob Dylan sounds like. . . he does.
Let me just be a little more precise, if I may.
Technically, you can play a diatonic chromatically, although it takes a lot of practice. Look at harmonica innovators like Howard Levy who do exactly this, by adding techniques such as overblows and overdraws to the standard arsenal of blows, draws, and bends. Also, you can get valved diatonics that make it easier.
As for playing in the wrong key, it depends on how you’re looking at it. Harmonica players will play the harmonica in a different key than the harp is listed, but I don’t consider that the wrong key. Folk harmonica (such as Bob Dylan) is played in 1st position aka “straight harp.” In a key of C harmonica, you’re playing the C major scale. Blues harp is most often played in 2nd position, aka “cross harp.” A C harmonica would be used to play G.
While most blues harp is played in second position, the example you mentioned, Little Walter, was quite famed for his third position playing.
However, there are positions beyond this. If you go up the circle of fifths, you’ll see that 3rd position (aka “slant” or “double cross” harp) would involve playing the C harp in D, fourth position in A, and so on. If you know music theory, you can see that essentially, 2nd position is G mixolydian (however, as you play blues, you often flatten the thirds with a bend and choose specific notes within the blues scale), 3rd position is dorian, 4th position is aolian (aka minor), and so on.
I think the OP’s head just exploded 
Maybe I should have put wrong in scare quotes. And apologies for repeating stuff you’d already mentioned (like chromatic harps, Bob Dylan even) I must have overlooked your first post.
The way I learned to play harmonica was that the holes were “fingered” on the mouth side with the tongue (great reason why you never lend/borrow a harmonica) My hands were never used to block the holes, only to muffle or supply tremelo to the sound. Changing the mouth opening size, mouth position, and tongue placement set the melody of the instrument.
Aha! One I can answer (SpouseO’s taking up the harmonica).
The diatonic harmonica that he plays has a series of holes - these are what you blow/draw through(drawing air across the reeds inside the holes) to make the notes. In the beginner’s books he’s using, you’re taught to use your lips so that you blow/draw through one hole at a time to produce only one note. If you don’t seal your lips properly for one note, you’ll blow through multiple holes at once, producing multiple tones for a chord-like effect. You use your hands to move the harmonica over your lips from note to note, or to produce certain effects. You don’t use your hands to produce the notes themselves.
Musicians who use rigs to support their harmonica so that their hands can do other things (such as strum a guitar or play a piano) need to move their head back and forth over the harmonica to produce the notes they want.
Yeah, definitely too much information for the OP. However, this being the Straight Dope and all… That’s what I love about this place. Nit picky and precise asides that I would edit out of regular conversation I get to use on the SD.
Usually, as somebody already said, notes are selected by moving the pursed lips and tongue across the holes.
However, a treasured few players, such as Norton Buffalo, form the tip of the tongue into a tube and dance that across the holes.
There’s actually a bunch of techniques for getting single notes out. The purse method is the most common, followed by tongue-blocking. Tongue-blocking gives you the flexibility of alternating single notes with chords (thus making your own rhythmic accompaniment). Lots of harmonica players use a combination of the two.
The third method is the one you mentioned, but a fourth, which I used at first before using lip-pursing, is to jam the harmonica into the flesh on the inside of your lower lip. It’s a bit easier done than explained, but basically you can form a little gap between the harmonica and your lower lip that’s precisely the right shape for playing single notes.
