First thing to remember about splash: There are four types of splash, radial, line, spell, and “special”. Of the four, special is limited only to Devourers, while radial and spell are the two most common ones.
Secondly, each splash has a certain area of effect (duh). Radial splash has splash originating from the origin in a circle-square pattern, which is basically what happens if you take a square are smooth out the edge a lot. It’s kind of like an oblong looking thing, with eccentricity closer to that of a circle. Line splash has splash in a line with a small radius of extra splash in the case of the Firebat.
Third, the radius of splash is varied. In radial splash, the epicenter of the attack will take full damage, while two other splash layers, deemed r1 and r2, deal 50% and 25% damage respectively. You can check Arsenal for exact ranges. The splash is dealt as a percentage of damage, after armour and damage type are applied. Certain spells such as Nuke and Irradiate have special instances of radii other than the two ranges listed in Arsenal (actually spell splash isn’t even given in statistics).
Ok now onto the different types.
Radial splash, which is used by units such as the Siege Tank, Reaver, and Infested Terran, deal damage in an epicenter of a certain radius (always small, but it will deal full damage to a splash), while damage radiates out in the two splash ranges, the r1 and r2 with the 50% and 25% damage. Some units such as the Siege Tank and Infested Terran deal friendly and allied splash damage, while Reavers deal only allied splash damage as well as splash to the enemy.
Radial splash also deals damage under Dark Swarm, but only the splash as a corresponding radius is dealt. As all attacks are shifted a bit when under Swarm (the pure sprite of the attack), the splash will be dealt at the attack target, while the epicenter of the attack sprite will actually deal full damage (not verified for 1.12, but it did for 1.08

The second type of splash, line splash, is damage dealt in a line from units such as the Lurker and the Firebat. This type deals full damage without a reduced-damage radius throughout its entire sprite. Thus, the Lurker will always deal 20 damage to any unit caught in its attack. A note on the Lurker is also the fact that the splash will actually carry farther beyond its actual range, so if you are targeting a Bunker that is out of range just slightly, if the Bunker is unloaded and you attack a Marine, the Bunker will take full damage.
A Hybrid of this is unique only to Firebats, which has a combo of both Radial or Line, or, depending on your perspective, radial only with a very elongated radius of effect. The Firebat deals full line damage in two short rows, but any units not caught directly in the attack animation will also take r1 damage only.
Spell splash is basically the area of effect for the spell, that is, you are either caught by the spell or not and take full damage or effects of it (exceptions are Nuke, and Irradiate). All units caught by the range of effect will take the effect.
Nukes give large r1 and r2 splash, and also have a tertiary splash range and is the only attack to do that (Other units that appear to have that effect such as the Archon are only taking r1 or r2 and receiving damage from the attack itself. It’s a strange effect. It’s also why Archon attacks sometimes seem to do more damage than they should).
Irradiate gives an epicenter of attack, but with stacked splash. It is the only splash in the game that has the potential to do more damage than the epicenter, as successive Irradiates will not do more damage (but renew its effect), but the splash will stack. What I mean is: if you cast 12 Irradiates on a Tank and Zerglings attack it, it will do the same damage as one Irradiate and one Irradiate's splash (the spell is on one target, and the effect is renewed). However, f you Irradiate 12 Vessels, Scourge will not be able to kill any as the Irradiate splash is in excess of 25 per second compounded through all the castings and is thus able to kill the Scourge. There are 12 Epicenters, and thus 12 splash radii which compound damage with each other at the splash only.
Special splash is limited to only the Devourer, which has a radius of 1 for its epicenter and will therefore only damage one unit, but its spell will affect neighboring enemies.
Of course many of you already know all of this, but I felt the need to contribute today

-yue
Edits: Minor formatting, made the post look prettier. Cleared up issue in semantics with the Irradiate explanation.