Half-Elf

A mage

Half-Elves are, (duh) the children of humans and elves. They maintain many of the characteristics their elvish and human parents, only watered down: For example, they live a wee bit longer than humans (they don't hit venerable until they're over 100 years old) but remain vigorous to the end like the elves. Typically the combination is human father/elven mother (for reasons outlined on the elf page), though notably the most famous half-elf; Arwen, had a human mother and an elf father.

Like half-orcs, enjoy a lot of the versatility of humans in Pathfinder, with a floating +2 bonus to any stat they like, free skill focus in any one skill, and the ability to select multiple classes as "favored" for the purposes of multiclassing, while getting elven bonuses to their Will saves and low-light vision. One particularly cool alternate ability lets them trade in skill focus to gain training in any one exotic weapon they like, just in case you always wanted to try an urumi or falcata but didn't wanna spring the feat for it.

A rogue

In 4e, they have a +2 Constitution (from their human parentage), and a choice of +2 Wis or +2 Cha (presumably from Elven parentage). They make shitty rangers, which is also bizarre considering they're descended from the one race that can do it right. They do make pretty amazing bards, and half-elf bards have a specific paragon path dedicated to them. On the other hand, they can warlord like a boss and they get a free encounter power from any class other than their own. If you like trawling through setting guides and back issues of magazines (or have CBLoader), half-elves are the class for you. The only downside is, you're playing a race with "elf" in the name, and nobody wants that.

In 5e, the dev team got out bottles and bottles of anabolic steroids. They are easily the most powerful race in the base book, and just for comparison, look at what they get compared to the poor dragonborn:

A +2 to Charisma, and two +1s in any two other stats, making them good at being Sorcerers and Bards, as they are the only race other than Tieflings who get a full bonus to Charisma. They also retain many elven traits. Also, they can grow beards, which is explicitly mentioned in the PH. This makes half-elves twice as manly as regular elves (though two times zero is still zero). The Sword Coast Adventurer's Guide lets you switch out the free pair of skill proficiencies for increased Elfy traits - do not take the Keen Senses option, as it's a load of bullshit (Darkvision + Perception proficiency, when you already got Darkvision + 2 free skill proficiencies before). You can, however, take a bunch of more interesting elf traits; the Elf Weapon Proficiency (which is free proficiency with some extra weapons), a 35ft base movement speed, a free cantrip from the Wizard spell list, the Drow Magic trait or, weirdly, a Swim speed of 30 feet to reflect Aquatic Elf ancestry. Despite no Aquatic Elf racial stats showing up in the game so far.

An interesting example of half-elven people are the race of Numenor in the Silmarilion, whose king was Elros Tar-Minyatur son of Eärendil and Elwing, both half elves. The Numenorians were characterized by their tall height, having extremely long lives and a judgment-impairing fear of death, being tricked by Sauron into warring with the gods for immortality (which got their home destroyed Atlantis-style), and are the race from whom Aragorn and the people of Gondor descended.

As you've probably seen by now on this page, they are also typically sexy ladies. This is not at all uncommon, and is probably some quirk in the breeding process.

Half-Elves in 5e

Just to give more concrete examples:

• Ability Score: +2 Charisma, +1 two ability scores of choice
• Typical Alignment: Favor any Chaotic alignment
• Size: Medium. Range from 5 to 6 feet tall.
• Speed: 30 feet walking.
• Darkvision
• Fey Ancestry
• Skill Versatility: Immediately gain proficiency with two skills of your choice. Can be swapped out with Sword Coast Adventurers Guide Variant rules for one Elven Trait.
• Languages: Common, Elvish, one bonus language

* SCAG Variant rules for half-elf provide rules for swapping your Skill Versatility for the following.
* A Half-Wood Elf can take Elf Weapon Training, Fleet of Foot, or Mask of the Wild.
* A Half-High Elf can take Elf Weapon Training, or the Free Cantrip
* A Half-Drow can take the Drow Magic feature.
* A Half-Aquatic Elf can take a swim speed of 30 feet.

A bard