Dispel Magic Only Ends Spells

A magical effect that is not the result of an ongoing spell cannot be dispelled.

Jeremy Crawford has issued several Twitter rulings to this effect.

In this Tweet:

Q: Can dispel magic undo a druid's wild shape

A: Dispel magic ends spells. Wild Shape isn't a spell.

and this one:

Q: Does Dispel Magic auto work vs Channel Divinity powers? Ie.. Paladin's Vow of Enmity magic effect.

A: Dispel magic ends spells. A Channel Divinity option like the paladin's Sacred Weapon isn't a spell.

and this series of Tweets:

Q: Does dispel magic effect an arcane ward?

A: Dispel magic ends spells. Arcane Ward isn't a spell

Q: Does that mean magical items cant be dispelled either? We've assumed that they could be disabled temporarily with dispel.

A: Dispel magic can end spells that come from a magic item, but it has no effect on the item itself.

Also note that Dispel Magic can only end spells with an ongoing duration. It cannot remove the lingering effects of an instantaneous spell. PHB P. 203:

Many spells are instantaneous. The spell harms, heals, creates, or alters a creature or an object in a way that can't be dispelled, because its magic exists only for an instant.

The March 2016 Sage Advice Column covers Dispel Magic as well. It reiterates that Dispel magic cannot end magical effects that are not the result of a spell, such as a vampire's Charm effect. It also reiterates the rule about Dispel Magic not working on the results of a spell with an instantaneous effect. It adds that a readied Dispel Magic cannot be used in place of a Counterspell, though in some limited circumstances, it may be almost as good.

To answer your actual question, in this context, targeting a magical effect clearly means you can target just one specific ongoing spell effect instead of stripping everything on a particular creature.

It depends on where they are standing

The darkness spell is described in such a way as to give the impression that it is something tangible that spills out around corners, but can be blocked by objects. Think of darkness as not unlike a fog cloud spell.

The darkness spreads around corners. [...] Completely covering the source of the darkness with an opaque object, such as a bowl or a helm, blocks the darkness.

Likewise, darkness engulfs non-magical light - or rather, it cannot be illuminated by non-magical light. This means that magical light can illuminate it.

Light in 5e is described as having bounds. Faerie fire indicates that afflicted creatures shed dim light, which only means that area imposes disadvantage on Wisdom (Perception) checks; attacking into dim light behaves as normal.

This is where location comes into play. The bounds of the dim light must penetrate the bounds of the darkness spell in order to be seen. Put another way, there must be no darkness between you and the creature. You need to check the positions and radius of the area of darkness and creatures affected by the faerie fire.

If the radius of the darkness spell's area completely engulfs the radius of the dim light, they cannot be seen by you from any angle. If they are 5 feet or more away from the center of the darkness, they can be seen from the direction the dim light is from the center. This is sort of mathy, but really no more difficult than normal line of sight. Darkness spills out around corners continuously, so just visualize or draw the circles and draw a line to the center of the dim light.

Who do I have advantage against?

Advantage and disadvantage hinge on who can see whom. In general, you have disadvantage on attacks against targets you cannot see, and advantage on attacks against targets who can't see you. However, when neither target can see each other (such as when a target is standing in darkness and you are not), you have neither disadvantage or disadvantage.

From the basic rules on advantage:

If circumstances cause a roll to have both advantage and disadvantage, you are considered to have neither of them, and you roll one d20. This is true even if multiple circumstances impose disadvantage and only one grants advantage or vice versa.

This means that faerie fire's advantage only has an effect when no disadvantage is imposed. It is important to note that, as explained in the question, faerie fire must be cast at a level greater than 2nd, or darkness will dispel it.

You have advantage against:

• Targets you can see that can't see you. It shouldn't matter the circumstances in which this takes place (Devil's Sight, location, etc); if you can determine who can see whom, this will suffice for advantage.
• Targets affected by faerie fire that you can see
• Targets in any situation in which you have at least 1 source of advantage and no sources of disadvantage

You have disadvantage against:

• Targets not affected by faerie fire that you can't see and that can see you
• Targets in any situation in which you have at least 1 source of disadvantage and no sources of advantage

You have neither advantage nor disadvantage against:

• Targets you can't see that also can't see you
• Targets affected by faerie fire that you can't see
• Targets standing in dim light that you have line of sight to (i.e. you can both see each other)
• Targets in any situation in which you have any number of sources of advantage and disadvantage (at least one of each)

This answer hinges on the fact that the darkness spell is dark in 3 dimensions, much like a 15-foot radius of fog or smoke. I believe this is true because of how the darkness spills around corners and can be blocked by objects.

If the darkness spell only "darkens" objects and ground in its radius, then it makes sense to imply that the faerie fire penetrates through it, making line of sight meaningless. Is it a floating black orb, or is it a radius of darkened objects? Again, I say the former, but I would personally leave this up to the DM.