3. And Ne'er the Two Shall Meet
OK, we can do light, and we can do darkness. But what if the two of them meet? There's quite a few interactions, including a special exception for the spell daylight, so first let's focus on the basic interactions. From the descriptors and the spells themselves, we glean the following facts: Spells with the light descriptor only raise the light level within an area of a darkness descriptor spell if they are higher level than the darkness descriptor spell. Apparently also, darkness spells can counter or dispel light spells of equal or lower level (and light spells can do the same to darkness spells). So what does that mean?
4. I Counter Your Counter!
There are many ways to misinterpret the "counter or dispel" text for light and darkness spells. Here's how that particular rule actually works. To counter a spell of the opposing descriptor, you ready an action just like any other counterspell. Just as normal for counterspell, the target of the spell must be within range (which, without Reach Spell metamagic, is touch for most light and darkness spells). If the target is in range, you automatically counter the opposing spell and it has no effect, just like always for counterspell. To dispel, you simply cast your spell on the same target (just like with enlarge person and reduce person) and then they cancel each other out, leaving no spell. Again, the range is usually touch and the target is the object that radiates the darkness or light; you can't just touch an arbitrary spot within the darkness or light.
5. Pierce the Darkness
Now that we have those out of the way, let's assume the more typical case where someone cast a darkness spell on one object, somebody else cast a light spell on another object, and the areas overlap. We're still not dealing with daylight yet. Based on the rules of light and darkness, here's how to adjudicate this situation within the overlap:
First, the darkness spell turns off nonmagical light sources and lowers the ambient light level. If there are multiple darkness spells, figure out the highest spell level (not caster level!)
Next, the light spells attempt to shine through. For every light spell, check to see if it has a higher spell level (not caster level!) than the highest spell level of any of the darkness spells. If so, that light spell has its normal effect, as per the spell. Do not reduce its light level again for the darkness spell; that already happened. This is true in all overlapping areas, as per the May 2013 FAQ, whether the light spell's source object is within the area of darkness or not.
6. Here I Stand, in the Light of Day
OK, so what about daylight? We've been putting that one off until now because it simply doesn't work like other light and darkness spells. As it says "Daylight brought into an area of magical darkness (or vice versa) is temporarily negated, so that the otherwise prevailing light conditions exist in the overlapping areas of effect." Daylight comes in, if necessary right after those last two bullet points in section 5:
If no other light spell is sufficient to overcome the darkness spells in the overlapping area, and if there is a daylight spell active in the overlapping area, the daylight spell's special negation clause kicks in (regardless of the spell level of daylight and the darkness spell; it just works, always). This means that you negate all the magical light changes in the area and bring it back to prevailing conditions. As a side effect of negating the magical darkness, those nonmagical light sources activate again (while they are not ambient, they were still part of prevailing conditions). Other magical light sources still are not active in the area; they had their chance to attempt to negate the darkness spells and didn't, so they were not part of the prevailing light conditions, instead subsumed by daylight's more powerful special negation clause.
Level 8 Feiya, Kyra, Ezren, and Damiel are traveling through the Darklands. Damiel brewed Ezren an infusion of darkvision, which he has active, as well as comprehend languages. Damiel also has low-light vision because he's an elf. The rest of the group had been relying on Kyra's heightened continual flame (heightened to spell level 4) to see, as well as various light cantrips, since the ambient light level is darkness. The group is ambushed by a group of darkfolk in a large cavern. In the first wave, dark creepers emerge from the darkness, each of them having cast darkness prior to the encounter. From the distance, no one, not even Ezren, can see the creepers, as they are beyond the range of his 60 foot darkvision. As they approach to 60 feet, Ezren spots them because of his darkvision. Damiel still can't see them because Kyra's heightened continual flame counts as extending 40 feet for the purpose of determining where it overlaps their darkness. As they approach within 40 feet, everyone can see them, as Kyra's heightened continual flame defeats the darkness in the area of overlap. The other light spells stop working, though.
Next, the dark stalkers advance, with their deeper darkness spells active. Even when they get to 60 feet, Ezren can't see them because it's supernatural darkness. However, since Kyra's heightened continual flame is heightened to 4th level, it keeps shining brightly. Since no one ever takes the Dark Folk language, the darkfolk use it to coordinate their attacks. Sadly for them, Ezren understands them anyway, and he warns Feiya that the darkfolk have a dark slayer who somehow heightened his spell-like ability deeper darkness to 4th level once per day through numerous blood sacrifices. Feiya nods, pulls out her rod of lesser reach metamagic and readies an action to counterspell with wandering star motes (which now has a range of 180 feet). Since wandering star motes is a 4th level light spell, the heightened deeper darkness is equal or lower level, so the counterspell ruins the dark slayer's big chance! The dark slayer snarls in anger and sends in its last big wildcard, a dark creeper barbarian, who sunders Kyra's heightened continual flame. This allows all those deeper darkness spells to defeat the remaining light sources easily, plunging the entire area into supernatural darkness, much to the darkfolks' delight.
Ezren ends their victory cheers early by casting daylight on his cane, which negates everything in the overlapping area, leaving the fight at the prevailing light level, normal darkness (hey, at least Ezren can see now!). Damiel, alchemist that he is, cracks a sunrod, which now provides light to everyone else. Desperate now, the dark slayer sends in the dark stalkers, who cast deeper darkness and then deliver the touch spell to Ezren's cane. They succeed, which dispels the daylight because daylight is equal or lower spell level.
Fed up with the whole situation, Kyra uses her 8th level sun domain ability nimbus of light, which instantly dispels all the darkness spells in 30 feet and then shines like a daylight. The battle is over soon after.