Terminology

AoE	AoE damage is short for "area of effect" damage. Generally speaking, AoE damage is damage that is dealt to more than a single target.
Board Clear	"Board clears" are effects that clear the opponent's board of minions. Normally, these are AoE damage spells, but clearing the board can also be done through a series of minion trades that eliminate all of the opponent's minions.
Board Control	Board control refers to the situation where a player has dominance over the game board. Generally, this refers to situations where one player has one or more minions on the board, whereas their opponent has none. Maintaining board control is extremely important, as it allows players to build devastating minion combinations.
Character	The term character encompasses both minions and heroes. If an ability deals damage to "all characters", it will damage all minions and all heroes.
Card Advantage	Card advantage refers to the situation when one player (the advantaged one) has more cards in their hand than the opponent. Generally speaking, having card advantage is a beneficial state, since it gives players more options for dealing with each situation.
Coining	The action of using The Coin, the extra card given to the player who goes second.
Draft	In Arena mode, it represents the 90 cards you are able to choose from to create your deck. The process of choosing cards is called drafting.
Drawing Power	Drawing refers to drawing cards from your deck into your hand. Drawing power refers to your deck's ability to maintain a steady flow of cards into your hand. Generally, high drawing power is good, since it gives players more options for dealing with each situation.
Drop	Drop is another term for "minion". You will hear this often in relation to the cost of the minions, such as "this is one of the best 2-drops in the game" (that is to say, one of the best 2-cost minions in the game).
Early Game	The early game refers to roughly the first 3-4 turns of a game.
Enemy	The term enemy refers to characters that do not belong to you. Some abilities are only usable against enemy characters, meaning that they cannot affect your own characters.
Favourable trading	A situation where a less expensive or important minion kills a more expensive or more important minion, and dies in the process;
Friendly	The term friendly refers to characters that belong to you. Some abilities are only usable on friendly characters, meaning that you cannot affect enemy characters.
Going first/going second	The player who goes first is the one whose turn comes up first at the start of the game. The player going second receives The Coin, and an extra card as compensation for starting second.
Hero	The term hero refers to heroes (such as Jaina Proudmoore, the Mage hero, or Garrosh Hellscream, the Warrior hero), and it does not include minions. If an ability deals damage to "the hero", it cannot be used to damage minions.
Late Game	The late game refers to turns 9 and beyond of a game.
Mana Curve	The mana curve of a deck refers to the distribution of cards within a deck, when taking into account their costs. A deck with a good mana curve is one that maximises the chances that on each given turn, you can play cards that have a cost that is close or equal to your maximum amount of Mana Crystals. For example, a deck that has only creatures that cost 5 or more Mana Crystals cannot play any cards for the first 5 turns of the game, which is a disadvantage. Likewise, a deck that has only low-costing cards will find that these cards can no longer compete with the opponent's high-costing cards late in the game (even if several low-costing cards are played together).
Meta-game	The term meta-game has two different meanings. It refers to matters of strategy that go beyond what happens in the confines of a single game between two opponents. For example, the general balance between classes (class X being considered more powerful overall than class Y), or the particular power of a specific deck for a class are considered matters pertaining to the meta-game. It also refers to matters of overall strategy within a single game between two opponents, but matters which are not strictly related to the minions and cards currently in play. For example, the desired outcome of the late-game, or the maintaining of card superiority can be considered matters pertaining to the meta-game, in a restricted sense.
Mid Game	The mid game refers to turns 4-8 (approximately) of a game.
Minion	The term minion refers to the minions present on the board, and it does not include heroes. If an ability deals damage to "all minions", it will damage minions, but not heroes.
MMR	Match-making rating; this is a hidden value that is attributed to each player that participates in ranked Play games. It effectively determines the "skill" of the player, and players are matched against one another according to their MMR. There is no way to see what the MMR of a player is.
Mulligan	To mulligan a card means to replace it, specifically at the start of the game, when you are given the option to replace some or all of your starting hand.
Over-extend	Over-extending refers to a situation where a player puts more minions down on the board than they need to, exposing these minions to forms of mass-removal. Generally, we talk about over-extending when the same result (keeping control of the board) could have been achieved with fewer minions.
Owner	The owner of a minion is the player under whose control the minion is in.
Play	A play is an action or a set of actions in a given situation. For example, there might be different combinations of minions and spells for you to use in a certain circumstance (considering the minions on the board and cards in your hand), and each combination would be considered a different play. You may also read this with the "good" or "bad" qualifiers. A good play is a play that was efficient in neutralizing a threat or giving you board control, for example, while a bad play might be one where you failed to make use of the best possible interactions available to you.
RNG	RNG stands for Random number generation. In Hearthstone, this refers to the situations where an outcome is decided randomly
Rush	A rush deck or strategy refers to a deck or strategy that focuses on summoning many minions quickly and dealing a lot of damage before the opponent can react. Rushes are usually countered by Taunt minions, or by having their flow disrupted (by board clears).
Synergy	Synergy is a term that refers to two or more minions or spells having a beneficial interaction that increases their individual values. For example, a Windfury buff has synergy with a buff that increases damage, since the damage increase is twice as great given the fact that the minion can attack twice. The same damage-increasing buff would be much less useful on a minion with which such a synergy does not exist. Generally, it is a good idea for most of the cards in your deck to have strong synergy between them.
Targeting	Targeting in Hearthstone refers to the action of a player using their mouse to select a minion to be the recipient of a spell or attack. You will mostly encounter this term in the context of minions which are exempt from being targeted.td>
Topdeck	Topdecking is used in a few closely related, but slightly different ways. Generally speaking, to topdeck means to draw from your deck.
Trading	Trading refers to a situation where a minion attacks another minion, and they both die as a result;
Vanilla	Vanilla is a term that refers to a minion that has no special abilities whatsoever (no Battlecry, Deathrattle, passive ability, and so on.
Zerg	See rush.