This one's a bit rudimentary, but scale length is a bit of a mystery to me. I understand what it is, but I don't understand what it does. I own guitars with different scale lengths (Telecaster vs Firebird vs Les Paul), but I have never really noticed a difference between them that I could attribute to the scale length. How does scale length effect my playing, or does it at all?
For those who don't know: scale length is the length of the string from the bridge saddle to the nut: the two points of contact that define the portion of the string that vibrates to produce the sound (technically, the vibration is transmitted through to the rest of the string, but for the purposes of this discussion, we can ignore that).
As for how it affects your playing: in a couple of ways.
The scale length of a Stratocaster is 25.5"; the scale length of a Les Paul is 24.75". Imagine you took a Strat in standard tuning and capo'd it at the first fret. The capo acts more-or-less like a new nut, shortening the effective scale length of the strings to roughly that of a Les Paul. But it also raises the pitches of the strings a half-step. So to get it back into standard tuning, you detune all the strings a half-step. So the shorter scale length has caused you to reduce the tension on the strings.
This relationship, of course, goes both ways. For a given string to be tuned to a given pitch, a longer scale length will require more string tension while a shorter scale length will require less. In practical terms, it means that if you put the same strings on a Strat and a Les Paul and tune them the same, the strings on the Les Paul will be easier to bend.
Looser isn't always better, though. 7-string guitarists and 5-string bass players have long noticed that the low-B string feels "floppy" or "mushy"---this is no coincidence. It's a direct result of the fact that the scale length of most guitars and most basses is too short for the low-B to have enough string tension to feel good.
The tighter the string, the more it produces higher overtones. This means that tighter strings will sound brighter and have more definition, while looser strings will sound warmer and rounder.
IMHO, in the long discussion of why Strats sound different than Les Pauls, scale length is often an overlooked factor. Single-coil vs. humbucker, alder/ash vs. mahogony, bolt-on vs. set-neck---sure, all of these make a difference. But a Strat is also almost a full fret longer than a Les Paul, and that's a big reason why Strat's sound so clear and bell-like on the bottom end, perhaps harsh on the top end, while Les Paul's sound mushy and indistinct on the low end but warm and round on the top. It's also why sticking a PAF humbucker on a Strat and gluing the neck to the body still won't make it sound like a Les Paul.
As I mentioned earlier, 7-string guitarists and 5-string bassists have struggled with the low-B string since these instruments were first developed. 6-string bassists have it the worst, since the high-C string is almost the same pitch as the D-string on a guitar, yet is almost 10" longer and also has a heavier gauge to boot. Most 4-string basses have a 34" scale length, while many 5- and 6-string basses have a 35" scale length to give the low-B string more tension and definition, but really, a 35" 6-string bass is the worst of both worlds.
I've seen string sets for 7-string jazz guitarists that use flat-wound strings for the E, A, D, and G strings, but a round-wound for the low-B in an effort to produce more definition. To me, this seems like an admission of defeat.
Pianos are shaped the way they are for a reason: bass strings simply have to be longer than treble strings. Violins, violas, mandolins, cellos, double-basses: these all have a range of an octave-and-a-sixth between their highest and their lowest strings. The truth is that a guitar, with a range of two octaves between its lowest open string and its highest, is pushing the limit of what a uniform-scale-length string instrument can get away with. Extended-range guitars and basses really don't work all that well (sorry! I love them too!).
Fanned-Fret or Multi-Scale Instruments
But what if you could design an instrument so that the bass strings were longer than the treble strings? Then the bridge and the nut could no longer be parallel, and neither could the frets, so instead you get a fret layout that looks like a Chinese fan. But it works! To give just one example: a Dingwall 6-string bass's scale lengths go from 37" on the low-B to 33.25" on the high-C. It looks goofy, but it's totally playable and sounds great.
Once upon a time, I designed an eight-string guitar (turns out, so have many others before me) with a low-A and a high-A. And as it turns out, there's just no way to make that instrument practical without fanning the frets. Otherwise the low-A sounds like mush and the high-A both sounds harsh and breaks all the time. But the fanned-fret instrument sounds amazing and has unbelievable range: it's almost piano-like.
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In my experience, scale length translates to increased string tension for longer necks.
This is a result of the physics of string vibration. Longer strings have to be at higher tension to have an equivalent pitch.
This is my own personal opinion, but I prefer shorter scale necks. Some people think they just "gotta have that extra two frets", but it only equates to a whole step. I come from the "Gibson Les Pauls are played by God" generation, and love the feel of the shorter neck. Even my Ibanez strat, which is my favorite guitar now-days, is 22 frets. I have two 24 fret guitars and their extra tension is just enough to throw me off until I reacquaint myself to the feeling.
And, yeah, a short-scale neck could have 24 frets, just as a long-scale could have 22, but generally the longer neck and more frets seem to go hand-in-hand.
Muscle memory, and practice should quickly negate any difference between the two sizes, so don't let it stop you... unless you're an anal old-school guy like me. :-)
The one factor that is definitely mentioned implicitly in the other answers, but not made explicit: If you would like to use lower tunings like dropped D or maybe even dropped C, you might want to opt for larger scale lengths. In the bigger scheme of things, this is of course why a bass has a larger scale length than a guitar, a double bass than a violin, etcetera. Also remember the larger scale lengths on baritone models. So I'm kind of stating the obvious here, but it seems to be an aspect that is easily overlooked by some people.
A few years ago I switched to a guitar with 24.75" scale length after having played 25.5" scale length guitars for a couple decades. I've noticed some differences. For one, the strings definitely feel more slinky. The slinkier feel, I think, also imparts a heavier tone. Think about it, so many great heavy guitar sounds are done on Gibson style guitars.
Switching scale lengths, I've found some positives as well as negatives. First off, I tend to play more melodic stuff on the shorted scale length. The guitar I'm talking about is an ESP LTD EC-1000. I'm fond of saying how that guitar changed my whole playing style, much for the better. Seems, though, that I feel what I'm playing much better on these guitars. Additionally, the looser feel allows me to raise my string action a good bit without making it hard to play. Doing this improves my tone exponentially! It's no secret that higher action usually can equal better tone.
One negative I've found, though, is with the strings as high as I like them, I sometimes run into intonation problems, because the strings are high enough that fretting them adds just enough length to the string to cause the issue. Slight, but still there. I've found that if I lower the action slightly lower than I like, I get the best of both worlds.....