Aggro decks have a Plan A - attack the enemy hero’s face - and no Plan B. Aggro relies on converting cards into damage through creatures and burn spells, aiming to end the game quickly. Rage rush is the prototypical aggro deck.
Aggro-control puts threats on the board and backs them up with protection and removal spells. These decks play enough creatures to kill their opponent (but not as many as full aggro) and try to protect those creatures with spells and effects. Aggro-control decks rely on maintaining tempo, playing proactively on the board, forcing an opponent to react, and maintaining board advantage.
Aggro-control sees more play in Conquest than ranked because, through bans, a player can tech the control aspects of their aggro-control list to expected matchups. Compared to a full aggro deck like rage rush, aggro-control tends to be stronger against aggro and midrange and weaker against full control.
Control decks aim to win in the long game. They generate card advantage, counter the opponent's threats, and at some point stabilize to achieve victory. Because of Spellweaver's unique 40-life win condition, some control decks play to win through healing.
Spread midrange decks aim to go "wide", overwhelming an opponent in the midgame while gaining card advantage. They are often enabled by Advanced Ne'eva's hero power, Reverence, which draws a card every time you play a creature. While Reverence no longer enables ridiculous late-game draw potential (with the power now restricted to the first three creatures played), the spread-midrange archetype still sees play, usually in one of three forms: nature/corruption "Hatebears", nature/order "Carebears," and rage/nature/order/wisdom "Rage Bears". (Sidenote: 'bears' is a term from Magic, referring to 2-drop creatures with 2/2 stats. Hatebears plays 2 mana creatures with negative effects, like death-curse shaman. Carebears plays 2-mana creatures with positive effects - like Cavalry Field Captain). Here's an example of a bears list that combines both ideas.
Value midrange attempts to hit a perfect curve, playing big creatures that out-value aggro and end the game before control can take board control or a combo can hit. Many value midrange decks will ramp levels or mana in the early game using cards like New Horizons or Angel-Blessed Knight. Value midrange may play control aspects in the early game so as to not fall behind aggro, and in that way these decks resemble aggro-control lists - but while aggro control lists will play early creatures and attempt to protect them (with possibly a few big creatures late), value midrange lists will play early creatures/spells so as to survive long enough to play a number of strong creatures on curve in the midgame. Alexa Valor is one of the stronger value midrange decks at the moment. Look how many strong cards there are in the 2 aspect/3 mana slot:
Mill decks try to force their opponent to run out of cards, and in a control-heavy metagame they can be effective. The most hated (or loved) mill deck is Snakes on an Azerplane. Here's the version Rinriet used to win his third title.
Combo decks aim to control the board and win in a single turn. They often get nerfed - because people hate losing to a combo they can't stop. Since nerfing the Dragon Combo, Banshee Combo, and Harbinger Train, we haven't seen a true combo deck in the meta. The closest is probably One-Turn Heal.