The Core Essence of D&D, To Me:
Characters
You play the role of a character (rarely more than one at a time) in an imaginary world. The world is a fantastic and quasi-medieval place, usually strongly influenced by Tolkien's writings of Middle Earth, where magic exists.
Characters are primarily defined by six attributes: Charisma, Constitution, Dexterity, Intelligence, Strength, Wisdom.
Each character is further defined by a race and class. The most typical races are dwarf, elf, halfling (aka hobbit), and human. The most typical classes are cleric, fighter, rogue, wizard.
Characters can improve over time. Their improvement is marked by experience points. After enough improvement they gain better ability scores and other mechanical effects.
The health of a character is defined by that character's hit points.
Mechanics
Characters have free-will in the game world and are generally not restricted to a following a certain path, restrained in their interactions with other characters, or made to or prevented from facing challenges at certain times as they would in a board game.
To resolve conflicts--essentially whenever a character attempts to do something that may fail or that is opposed by another character or hazard--players roll dice. Depending on the situation the dice rolled may have four, six, eight, ten, twelve, or twenty sides. The result of the roll may be modified by adding to or subtracting from its value, and success is determined by comparing that final result to a target number. The reason for rolling and the result of the role are usually fantastical, such as trying to magically hit a goblin with a ball of fire.
One of the players does not play a character but instead acts as the Dungeon Master, adjudicating the rules, arbitrating conflict, creating the game world and its hazards, and generally guiding the play of the game.
Hazards
The world is filled with mythical monsters, traps set by intelligent actors, and fantastic terrain.
Characters may explore vast underground caverns and dungeons, the unexplored wilderness of the game world, cities and towns of the land, the halls of government, or even the various planes of existence.
Rewards
Characters receive treasure as a reward for exploring their world and overcoming its hazards.
The most common reward is money, usually represented as coins of gold, silver, or copper. Monetary rewards may also include coins of different metals, jewelry, and art objects.
Rare but desired rewards are magic items, from mundane items and combat gear enhanced with mystical power to artifacts that have their own consciousness.