Hopefully I can keep this brief (not my style admittedly)...
Vf = Forward Voltage (Vf(min), Vf(max), Vf(nom) are maximum, minimum, and nominal abbreviations that I use, nominal is the typical factory rating of either batteries or LEDs in operation, DD is "direct-drive (off batteries).
The LEDs have a minimum voltage they'll operate at (and either plummet in output or simply won't light at all), below this no harm will come to them, they just won't work... they have a max they'll operate at SAFELY (due to heat buildup, beyond a certain point they'll literally fry internally, either the die, the epoxy, the epoxy, or everything...)... look at some of the datasheets and you'll find lots of 'derating' curves and such (so if you know an LED will be sealed in a hot environment, you can cut back other parameters to stay in the safe operating envelope of the LED.
Now that's the voltage range, however LEDs are actually "current driven" devices... the voltage is a by-product of the current they are operating at (as current increases, the voltage increases as well, but you can increase voltage and current might drop, depending on your source)...
So to optimize designs, you need either a voltage source that is in the operating range of the LED and can generate a suitable amount of current for operation (you can have a 3.2V LED, running off 3 NIMH cells (3.6V nominal) that are discharged, but still showing 3.2V, and the LED may not light up due to the internal resistance of the batteries preventing the current from flowing.
Most direct-drive (battery driven directly) LED lights are based on the nominal cell voltage falling within the LED ranges.
The biggest design problem with direct-drive is "thermal runaway"... an LEDs resistance (which isn't big to start with) drops as it's temperature increases (unlike most electrical parts).... if you run an LED near it's Vf(max), and it can't get rid of heat to ambient fast enough (enclosed, no heatsink, etc)... it'll start to heat up and the resistance to current will decrease... as this happens, the current increases, causing the heat to rise further, lather rinse repeat... the LED "runs-away" until you get a "friode" (fried LED)... these can short "closed", causing more current increase in neighboring LEDs, eventually blowing the entire set if it's an array.
Resistors in each voltage series helps avoid this for direct drive operation, as the resistance of a resistor increases with heat and higher current, it more than negates the drop from the LEDs, cancelling out this problem.
Regulators maintain a "constant" current to the LED array (array of 1 or more units)... in this regard, as the LED heats up, although it's resistance drops, the regulation circuit senses the change and adjusts to maintain a set current, essentially preventing any chance of runaway. If you have an array of parallel LEDs, thermal runaway is still possible, if run near limits and one starts to runaway, it'll take current away from other parallel legs, so to prevent this, small resistors can be added as well even in regulated circuits... however if designed reasonably within limits, this problem isn't likely to happen. Also, most regulators vary voltage to limit the current, and this will negate again thermal runaway (the Vf of each parallel leg is always the same, if an LEDs Vf starts to rise, it would require other Vfs in it's parallel leg to fall, but they must all share the current in that series, and act as "mini-resistors" to each others changes in Vf, and will basically "parasitically" take the current they need and thus stabilize the current, which will box in the misbehaving LED, so long as the regulator stays constant. It's not foolproof, and if you mix LEDs of significantly different Vfs, or push them right to the limits, stability can suffer and fail, but it's pretty reliable up to the operational and environmental limits. Exceed "mA max" and "Vf max" ratings at your own peril though.
Okay, the reasons regulators aren't used more often are numerous, however some of the most obvious...
1 - They take space, valuable especially in small AA lights
2 - They consume some power and drop efficiency:
A> Linears are (typically) 70-90% efficient (based pretty much on matching Vf of LEDs to Vin of batteries, the wider ranges needed (for instance, to accomodate alkaline and NiMH cells), the more it suffers
B> Switchmodes depend on types... buck/boost models (can handle a wide range of inputs above and below the output needed) are the least efficient but most flexible, typically 60-80%... boost models are typically 80-90%, but have to have the input a certain amount below the output, and buck models are the most efficient (90%+), but require the input to be some amount above the output
3 - They add complexity to design
4 - They add cost
Advantages are pretty obvious in most cases...
1 - Regulation - constant brightness over time until Vin drops below regulation capabilities
2 - Control options - most have brightness level, strobe, etc controls.
3 - Extra maintained brightness - These can be made to maintain higher brightness than direct driven lights, at the expense of runtime (using all available source power)... this can be a disadvantage also if a "cutoff" isn't programmed in, battery damage can result.
Design considerations...
Direct drive will give long-life but with steadily decreasing output... a resistor also needs to be added to prevent a too-high voltage input from blowing the LED, but the resistor will always be wasting some power in the form of heat as a result. Best designs put the Vin(nom) closest to the Vout(nom)... so if you have a 3.6V battery source (LiIon), and a 3.6V LED, it's a good match (and why you see so many LiIon flashlights on this website, as both are pretty much the nominal figures, so minimal to no resistors are needed).
If you got an abnormally strong LiIon battery (say 4.2V fully charged) and a LED with a real Vmax of 3.8 (kind of low for most power-white LEDs), you might blow it, and you'll see some people recommend slightly discharging the LiIons before using in certain flashlights for this reason (the 4.2V off a strong charge quickly drops below 4.1V when used for just a few minutes). A small resistor can protect against this, at an efficiency penalty.
Linear regulation will give safe regulation (too much Vin won't blow the LEDs), at the expense of efficiency, and runtime will be shorter than direct drive.
Switchmode regulation can be designed for many scenarios... long runtimes (using PWM or PFM switching to "dim" the LED for example can greatly extend life over either above method, especially in a buck converter), brightest output possible, brightest maintainable output for a given interval, etc... the simplest are nothing more than a pre-programmed current regulator though, if you have a 700mA LED that you want max rated brightness from for as long as the source can support it, buy/build a 700mA constant current regulator, linear ($2) for cheap, switchmode ($20+) for optimal efficiency.
Okay, not brief, hope that gives a quick 101 on the question.