The short answer, no class beats any other class 100% of the time. Decks can vary a lot even within one class.
The long answer, know the skills of every class. They all have things they are better and worse at doing. Here are some topics to think about how they rate: Card Draw, Tempo, Direct Damage, Control (minion kill), Control (AoE damage), Speed, Late Game finishers, Consistency.
Masters of Stalling. They have solid late game and a mix of heals and card draw. They make good counters to rush decks because of the amount of cost effective taunts available. Their weakest point is the first few turns, but that can be circumvented with Innervates and Neutral Minions. Their cards have a lot of "Choose option 1 or option 2" which make them more versatile than most other classes, so they have fewer weaknesses, but they are also not a direct counter to any class either. Druids tend to be more "Jack of all Trades, Master of None". Probably one of the most balanced classes, but that makes them tend to be more difficult to play as well because there are more choices.
Strong early game. They have potential for mid-game play with the change to Unleash the Hounds. They tend to be weaker in the late game but they tend to do well in arena using a Control archetype. (I went 2/3 and 3/3 my first two times with Aggro Hunter in arena and have gone 7/3 and 7/3 my last two times with Control Hunter). Hunter's AoE is not the strongest so they can die to swarm decks, and they also tend to be weaker to AoE due to the amount of cheap class-specific minions.
Masters of Direct Damage. They also have strong control in early game. The amount of direct damage they can due makes them a huge threat to Warlocks. Also very good vs rogues, paladins, and warriors who do damage to themselves with weapons to clear minions and do not have a lot of healing. When playing against a mage, do not over-commit to the board (do not play out your entire hand) because it is likely they will be able to wipe the board once or twice and you will need to refill.
Masters of minion efficiency. The ability to drop 1/1's all the time and use effects like Blessing of Might and Blessing of Kings to buff them for very cost effective trades helps to create card advantage. Additionally, Argent Protector can create huge tempo swings with his shield. Paladins tend to be more suseptible to AOE removal.
Masters of the late game. The ability to heal minions allows priests to start gaining card advantage on the board around turn 4. They have a lot of ability to draw cards and stall with their healing. They are weakest against fast decks. I would say one of the worst things to see is a rogue coin into Defias Ringleader turn 1 against you because of the amount of pressure you will be under from that point on.
Masters of Tempo. Rogues can push a lot of damage into an opponent's health early in the game and they have cost efficient direct damage. Rogues are weak against decks with strong early game control (like Mage) because they can easily stall out vs control.
While the randomness of their ability makes them one of the most unreliable, they do have some very solid removal cards. Their ability to play under-costed cards with Overload is both a blessing and a curse. A turn 5 earth elemental can be super effective if not answered, but if it is hit by a strong removal it becomes a huge tempo loss (Polymorph, Hex, Shadowword Death, Assassinate, Mind Control if late game, Naturalize, Sap). Shaman is more inconsistent than most other classes because with overload you are investing a lot more resources into each card. Bigger upsides and downsides to each play typically.
Masters of Tempo. Warlocks ability keeps them on top of the chart for Tempo. If they do not take a lot of damage early in the game, their ability will regularly win them games at the end. Warlocks are weak against any and most classes that can put out a lot of damage fast. Due to their lack of healing, they pretty much have to rely on being faster to kill their opponent (stalling is generally not an option).
Masters of Direct Damage. Not in the same way as the mage with spells, but warriors have the most charge and some of the best weapons for direct aggression. Their ability to generate armor also should be thought of like a heal, but better. They can draw a moderate amount of cards and use that to maintain tempo. Overall, quite well rounded.
Edit: I removed the link to Hearthstats because the link to the current month's report will continue to change each month.