Yes, they can see and attack with advantage
In summary, characters with darkvision can see in darkness as if it were dim light. Dimly lit areas do not block vision, so they can look right through the area with their normal, unlimited vision. And if their target cannot see them in the darkness, they get to attack it with advantage.
Darkness in the game
In the real world, darkness does not block your vision — if you stand in darkness at night, you can see a campfire in the distance just fine, even without darkvision. (Thanks to @DaleM for pointing this out). Darkness in 5e from a rules mechanics perspective however creates a heavily obscured area that blocks vision, p. 183 PHB:
A heavily obscured area--such as darkness, opaque fog, or dense foliage--blocks vision entirely.
It is treated just like opaque fog. On a strict reading, by the game rules you cannot see through even normal darkness as you could in the real world (for more discussion on how to adjucate this unexpected result, see Does the Darkness spell block vision?)1.
Here however, the character in darkness has darkvision. Darkvision says (p. 183 PHB):
Many creatures in the worlds of D&D, especially those that dwell underground, have darkvision. Within a specified range, a creature with darkvision can see in darkness as if the darkness were dim light, so areas of darkness are only lightly obscured as far as that creature is concerned.
So, for a creature with darkvision you treat squares within the radius as only lightly obscured, and a lightly obscured area of dim light does not block vision. Without heavy obscurement's unnatural rule, looking through dim light works just as in the real world.
Darkvision does not limit your other senses
Nothing in the description of darkvision says it restricts the creature's other senses in any way. Would a creature's other senses enable it to look out beyond? If the answer is yes, they can see beyond. So the elf can see creatures beyond 60 feet in clear daylight normally. The world does not appear dark to them beyond the radius of their darkvision, if it isn't.
If the creature they are attacking cannot see them in the darkness, they get advantage on their attacks against it, and the creature has disadvantage to attack them, because (p. 195, PHB):
When you attack a target that you can’t see, you have disadvantage on the attack roll. (...) When a creature can’t see you, you have advantage on attack rolls against it.
1Interestingly the DMG on p. 105 says, "Bright light in an environment of total darkness can be visible for miles.". This contradicts the effect heavy obscurement from darkness would have, because for this to work you need to be able to see light through darkness, like in the real world. If your DM rules like this, you would not even need Darkvision to see someone in a bright space outside the area of darkness you are in. Kudos to @Picklespeare for finding this.