Tempo ends up having several meanings - though in essence, the general definition would refer to the “momentum” or “change of board state” during a match. If you play out 4 minions on your turn, you are swinging the tempo or momentum significantly in your favour. You have the tempo because you have more mana worth of minions on the board than your opponent.
If you have that in mind - then you have an idea of what other phrases mean when combined with tempo:
a Tempo play, or tempo’ing out - commonly refers to playing out a minion for its stats. It has increased relevance when, you choose to “tempo out” a minion without gaining benefit from its Battlecry: eg: “tempo’ing out” Captain Greenskin for its 5/4 stats without a weapon. Or playing Rotnest Drake (6/5) without having a Dragon in hand or enemy minion to destroy. This is opposed to keeping the card in your hand for value, to play later. You may need to gain some presence or “tempo” on the board this turn, and have few other suitable plays, so you cannot keep that card for value.
A card is considered “low tempo” if it does not put stats on the board corresponding to its cost. Eg: Arcane Intellect is 3 mana that draws 2 cards. You are expecting a 3/3 or better for 3 mana for it to be considered a good tempo play
A tempo swing is a play or series of plays that usually results in a player gaining or losing a significant board advantage (tempo gain or tempo loss).
Cabal Shadow Priest can be 1 card that results in a tempo swing by itself. For 6 mana, you get a 4/5 and take an enemy minion with 2 or less attack and bring it to your side. Say you take the opponent’s Bone Wraith. They just lost 4 mana worth of tempo and you gain 4/5 (4 mana worth) + 4 mana Bone Wraith: 8 mana worth of tempo, for a 12 mana worth of tempo swing in total.
Combinations of Lackeys, and removal such as Eviscerate, Backstab, Sap, can provide high tempo swings in a single turn - so Rogue is particularly known for this.
A tempo deck in Hearthstone is a deck that often aims to play out cheap minions in the early game to take the board, and have tools to keep the board afterwards. In other words, “tempo out” cheap minions and continue to keep/regain the tempo. Galakrond Rogue is a tempo deck, previously mentioned with its cheap minions like Pharoah Cat and EVIL Miscreant generating Lackeys, and cheap removal to keep the tempo.
Demon Hunter is also a tempo deck with Battlefiend and Satyr Overseer providing very high tempo for a low cost, when used with the hero power. For 2 mana, Battlefiend is a 2/2 and you deal 1 damage. For 4 mana, Satyr Overseer is a 6/4 split across 2 minions and you deal 1 damage. Both cards have the ability to grow and provide more tempo+value in future turns. You might also called it an aggro deck, as it deals damage quickly and looks to end games.
Tempo decks rarely run expensive removal like Chaos Nova, because those cards are a tempo loss.
The difference between a tempo deck and a midrange deck is that midrange decks have more beefy minions, especially at the 4-5 mana cost. They are usually not as flexible as tempo decks in terms of cheap removal and tempo swing turns, rather they deal with threats by using their minions to trade instead.
Dragon Hunter is a midrange deck, and you will see a host of 4-5 cost minions: Evasive Feywing, Big Ol’ Whelp, Rotnest Drake. Stuff you play “on curve”, and in the case of Rotnest Drake, can provide an additional tempo swing.
So as you can see, tempo as a word can be used in many contexts and have different meanings that are connected to each other.