One of the key concepts that needs to be learned for practically any CCG is the idea of card advantage. The super basic idea is that you want to trade one of your cards for two of their cards as often as you can. Since each player draws one card per turn if you can make more of those trades than your opponent does then eventually they'll have no cards and you'll have a couple left to use to actually win the game. It's a lot more complicated than that in reality, and especially in Hearthstone in particular, but it's still a very important thing to keep in mind.
There are some cards that just innately generate card advantage without either player having much say in the matter. Mage has a card 'arcane intellect' that costs 3 and draws 2 cards. There's not anything that either player can do about it... If the mage has time to squeeze that spell into their rotation they're just going to go up a card. Similarly, azure drake is a 4/4 for 5 that draws you a card when you play it. That's innate card advantage assuming you can get any benefit out of the 4/4 body (and you almost certainly will!) There are also weapons like the paladin's truesilver champion that basically guarantees to kill 2 of the opponent's 4 toughness creatures. You can work to minimize the value they get out of the second attack by using taunts or selectively choosing which creatures to put in play but unless you have one of the few cards that kills a weapon they will straight up get to 2 for 1 you just by virtue of drawing the weapon.
On the other hand there are quite a number of cards that may or may not trade for multiple cards, and the play of both players decides just how good they are. The biggest example here is the mage spell flamestrike. It costs 7 and does 4 damage to every creature your opponent controls. If your opponent has a single 8/8 in play then it doesn't even get card parity. If they have a full 7 creatures in play all with 4 or fewer toughness then you get a massive 7 for 1! Both players get to work to make the flamestrike better or worse with the choices they make. The mage can use freezes and stealth creatures to limit the trades their opponent can make in the turns leading up to the flamestrike, letting their opponent potentially overextend into it. The opponent can proactively make trades instead of going face in the turn leading up to 7 mana being available. They can hold onto sticky/big creatures and play them before the flamestrike to minimize the damage done and keep stuff on board after the flamestrike.
Now, if you want to properly play around these kinds of cards you need to know what they all are. This is true in both limited and constructed, with a higher priority in limited being placed on commons and a higher priority in constructed being placed on the more powerful ones regardless of rarity. Anyway, I wanted to make a list of potential card advantage removal spells that can be played around. And then Matt asked me if I had one, so I figured I really should make one. So here we go! Broken down by class and rarity after the break...
Poison Seeds - 4 - Destroy all minions and replace them with 2/2s. Countered with creatures that have deathrattles, or creatures worse than 2/2, or not overextending all your big creatures at once. Is pretty bad, unless comboed with starcall in constructed, so probably not worth thinking about.
Swipe - 4 - 4 damage to one enemy, 1 damage to all other enemies. Countered by working to prevent trades that leave multiple creatures with only 1 health, or by playing creatures with 5+ health when you have 1 health creatures. Incredibly powerful and definitely important to keep in mind in both formats.
Starfall - 5 - 5 damage to a creature or 2 damage to all enemy creatures. Countered by working to prevent trades that leave multiple creatures with 2 or fewer health. Both aspects of this card are overcosted a little because of the flexibility provided, but is still very good.
Explosive trap - 2 - Triggers when hero is attacked, does 2 damage to all enemies. Countered by only attacking the enemy hero when a trap is up if you're willing to take 2 to all your things. You can trade off weaker things first to limit the impact. You can also probe for other traps to find out if it's more or less likely to be explosive trap.
Unleash the hounds - 3 - Get a 1/1 haste for every creature the opponent controls. Countered by trading aggressively to minimize your own creatures in play, or possibly even by holding back unnecessary creatures if the extra 1/1 would be crippling?
Multi-shot - 4 - Deal 3 damage to two random enemy minions. Countered by only having 1 creature in play, or by having tons of creatures in play, or by minimizing the creatures below 4 health.
Misdirection trap - 2 - Triggers when hero is attacked, makes the attacking creature attack something else instead. If you'll be sending multiple creatures at the opponent's face you want to send the weakest in first to minimize the chance you lose two creatures to one spell.
Explosive shot - 5 - 5 damage to a minion and 2 damage to the adjacent minions. Countered by paying attention to where you play creatures to minimize the potential threat. (Bigger creatures on the outside, or stealth/untargetables on the inside.)
Arcane missiles - 1 - 3 damage distributed randomly among enemies.
Arcane explosion - 2 - 1 damage to all enemy creatures.
Cone of cold - 4 - Freeze a minion, and the adjacent minions, and do 1 damage to them. Potentially countered with proper creature placement.
Flamestrike - 7 - 4 damage to all enemy creatures.
Blizzard - 6 - Freeze all enemy minions and do 2 damage to all of them.
Consecrate - 4 - 2 damage to all enemies.
Equality - 2 - All minions health gets changed to 1. Mostly used in constructed in combination with consecrate as a board clear.
Avenging Wrath - 6 - 8 damage distributed randomly among enemies.
Circle of healing - 0 - Heals all minions for 4, but there's a 4 drop rare priest minion that turns healing into damage.
Holy nova - 5 - 2 damage to all enemies, 2 healing to all friends.
Mind control - 10 - Steal a minion permanently; it doesn't get to attack immediately.
Shadow madness - 4 - Steal a minion with 3 or less attack for the turn; it gets to attack.
Lightbomb - 6 - Deals damage to each creature equal to its power.
Cabal shadow priest - 6 - Steal a minion with 2 or less attack permanently; it doesn't get to attack immediately.
Betrayal - 2 - Make an enemy minion deal damage equal to its power to the adjacent creatures.
Fan of knives - 3 - 1 damage to all enemies and draw a card.
Blade flurry - 2 - Destroy the rogue's weapon and deal damage equal to its attack to all enemies.
Forked lightning - 1+2 - Deal 2 damage to 2 random enemy minions.
Lightning storm - 3+2 - Deal 2 or 3 damage to all enemy minions.
Hellfire - 4 - Deal 3 damage to absolutely everything.
Dread infernal - 6 - Deal 1 damage to everything except the infernal itself (which is a 6/6).
Shadowflame - 4 - Destroy a friendly minion and deal its attack damage to all enemies.
Twisting nether - 8 - Destroy all minions.
Whirlwind - 1- 1 damage to all minions.
Cleave - 2 - 2 damage to 2 random enemy minions.
Brawl - 5 - Destroy all minions except one chosen at random.
Explosive sheep - 2 - 2 damage to all minions when it dies. (1/1)
Mad bomber - 2 - 3 damage randomly split between all other characters. (3/2)
Unstable ghoul - 2 - 1 damage to all minions when it dies. (1/3 taunt)
Knife juggler - 2 - 1 damage at random each time you summon a minion. Comboes with some class cards that generate a lot of creatures at once, particular the hunter 'unleash the hounds' and the paladin 'muster for battle' cards. (3/2)
Wild pyromancer - 2 - 1 damage to all minions when you cast a spell. (3/2)
Mind control tech - 3 - Steals a random enemy minion if they have 4 or more when it enters play. (3/3)
Abomination - 5 - 2 damage to all characters when it dies. (4/4 taunt)
Madder bomber - 5 - 6 damage randomly split between all other characters. (5/4)
Doomsayer - 2 - Destroy all minions at the start of your turn. (0/7)
Baron geddon - 7 - 2 damage to all other characters at the end of your turn. (7/5)
Foe reaper 4000 - 8 - Deals combat damage to adjacent creatures too. (6/9)
Deathwing - 10 - When he comes into play discard your hand and kill all other minions. (12/12)