I dont understand Quests

I played AI, that doesnt complete quests.
I played “standard hearthstone experience” with match making against human people, that doesnt complete quests
Battle grounds you can’t play with “A deck” you get random stuff, so that wont complete quests.
mercenaries doesnt use a deck, so that doesnt complete quests

How exactly am I meant to complete daily or weekly quests if nothing counts towards it? Help. Does it only apply to the super serious Ranked which I assume is found somewhere like Arena?

Correct; unless the quest itself says differently, quests can only be completed in PvP.

Er, it should. Can you give specifics? Exactly what quest did you have, and exactly what did you play, and what happened that you believe should have completed the quest? (“Standard hearthstone experience” is pretty vague) Just playing doesn’t complete quests, you have to do what the quest says. Also keep in mind that if you start a game and then immediately resign, Hearthstone won’t count it towards a quest (say you had a quest to play 3 games as a set of three classes including Mage; starting a game as Mage and then immediately resigning won’t count as a game played as Mage for the quest). Based on the particulars, you may have grounds for a bug report.

Not all quests require you to have “a deck”. Some quests can be completed in Battlegrounds–indeed, there’s some that can only be completed there. Mercenaries play doesn’t complete quests; Blizzard has stated this.

To elaborate: quests which say to destroy minions, deal damage to heroes or minions, and other things which can happen in BGs will definitely complete there. Some are much easier. For instance, it’s faster to deal 200 damage to enemy heroes in BGs than constructed.

I had a quest not too long ago that called for me to finish in the top 4 in a round of Battlegrounds–by definition, that quest could only be completed in Battlegrounds.

Blizzard must have changed Mercenaries in that regard, because that’s the mode I’ve been primarily using to complete quests. I just finished a quest to deal 100 arcane damage, and another to kill 30 minions, both done in Mercenaries while farming bounties and coins.