Which Hearthstone class is the best?

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Each Hearthstone class is unique. Most of them have their own mechanics, such as Shaman's Overload or Rogue's Combo, while some have their own flavour to them, like Hunter and his beasts or Paladin and minion buffs. This, however, poses a problem for game's balance: how to make each class unique, yet equally strong? The answer is sadly left unanswered, and some classes are simply better than others, While, each class shines in some places or under some circumstances, some parts of the game simply do not matter as much; the main focus of Hearthstone is on Constructed and Arena, where some classes are simply better than others. However, this is meta-dependent, so we will provide a mostly general answer, not tailored for current meta to ensure that it is correct even after the meta changes.

The strongest classes for Constructed often change, as the meta is quite unstable due to frequent balance changes and new sets being released. Generally, a class is considered to be the strongest when they have more high-tier decks than any other class. Strong Basic and Classic class cards are a great step forward to being the strongest class, and for players on a very tight budget, this (coupled with a strong Hero Power) often makes the class the best out there. An example of such class is Mage, who has great Basic cards, like Fireball, which makes it a good class for new players, and Classic cards, like Mana Wyrm, which allows the player to slowly transition into a more powerful deck even with low dust total.
For players with a higher budget, Hunter is a strong choice as aggro and midrange decks for that class have been here since the game's inception and continue to be a force to be reckoned with.

In Arena, classes rotate in power as well, but not to the extent of Constructed. This is due to the fact that each class has its own Hero Power, which is consistently present in each Arena playthrough, and therefore can be relied upon, unlike cards which are dependent on luck. For a Hero Powers to be considered powerful, it needs to affect the board state and be efficient and reliable in doing so. This makes Rogue's Hero Power strong, as the life total does not matter as much in arena, as one cannot consistently draft an aggressive deck or pack enough burn spells to finish the opponent off from mid-range life total.
Mage has also a great Hero Power, because unlike Rogue's Wicked Knife, it can damage minions and Heroes hiding behind Taunts. This is further complemented by Mage's strong Basic and Classic class cards, which often results in the class' dominance in arena.