Yeah. There's vaguely defined flavors, but since cross-classing is really only just coming to be a thing with copying hero powers and spellslinging and everybody getting to be a paladin, it's not really thoroughly fleshed out.
Like penguinofhonor said, Mage gets the most premium spells and awkward play-forcing tools. Thematically, it casts spells which make the boardstate difficult to gain an advantage on for your opponent, and then slowly chips an opponent down until it can finish them with cards like fireball or pyroblast.
Priest has health manipulation tools, high face damage spells, and sneaky stealing everything you do for whatever reason. Thematically, it has high health minions which it plays and then trades and heals in order to get board control as well as taking all of the enemy's stuff from them.
Druid has high quality minions, mana advantage cards, and proactive actions, but horrible reactive action (AoE, single target removal, etc). Thematically, it plays cards which give it extra mana to play expensive, high flexibility cards which allow it to control the board through tempo, until it hits the late game and just overruns the enemy through having a heavier mana curve deck.
Warlock has mediocre class cards except for powerful demons due to having the best hero power by a mile, but has the best board wipe options. Thematically, it abuses its ability to draw cards freely to gain a significant amount of card advantage and overwhelm the opponent with having more cards to throw at them, while slowly killing itself.
Hunter has generally cheap damage, and beast synergy cards. Thematically, it tries to aggressively force unfavorable trades for the opponent by abusing their health pool directly and killing them before you run out of cards to throw at them, while extending later into the game with powerful beasts and beast-support cards.
Warrior has self-healing galore, weapons to encourage hitting stuff with your face, chip damage aoes and charging stuff. Thematically, it armors up and directly face-tanks opponent minions with constant weapons, following up with charging minions into the opponent's face to finish them off, catching up on the board by killing everything together.
Rogue has incredibly efficient cheap removal, returning cards to hand, and insane weapon manipulation tools. Thematically, it plays many cards together to combo them and go from unwinnable board states to unloseable, very rapidly.
Shaman has cards which are insanely powerful in a vacuum but synergize poorly at best, and ridiculous amounts of spell burst. Thematically, it plays incredibly high quality cards ahead of time with overload and gaining tempo which they coast into victory with several unexpected spells doing an absurd amount of damage from hand.
Paladin has token spam, minion buffs, and reactive healing. Thematically, it plays a bunch of small minions and then makes them stronger, while protecting them with spells and hitting opposing minoins, and healing off the damage.