What is Corrupt in Hearthstone?
Hearthstone has many card effects that are common enough that they get their own keyword - a word or two that immediately tells you how the card works. Let's take a look at the Corrupt keyword.
Upgrades in hand after you play a card that costs more.
Corrupt is a mechanic in which cards with Corrupt will upgrade themselves in the manner specified on the textbox while they are in your hand after you play a card that costs more than the Corrupt card itself. Corrupt cards have "Corrupt" on their textbox when in their base form, and "Corrupted" on their textbox when in their upgraded form. Corrupt cards only need to be upgraded once before they reach their most powerful form. This mechanic is exclusive to Madness at the Darkmoon Faire.
Keywords that Work Great with Corrupt
Corrupt by itself has no keywords that directly synergize with it, although Corrupt cards benefit from a controlling game-state in which you can survive long enough to use them in their upgraded form. Any effects that help you control the board and stay alive (Taunt, Rush, Lifesteal, etc) will be useful here.
Twinspell cards like Unleash the Beast and The Forest's Aid can allow one card to have two separate instances of upgrading Corrupt cards.
Playing Against Corrupt in Hearthstone
While playing against a player that you suspect might have a Corrupt card in their deck, two things to take note of are cards in their hand they've been holding for a long time, and the Cost of cards they play. If your opponent plays a high-Cost card, then you may suspect that they've upgraded a Corrupt card in their hand, especially if that card in particular has been in their hand for several turns.
If you notice that your opponent has played a lot of Corrupt cards over the course of one game, then they will likely will have Y'Shaarj, the Defiler in their deck. In this case, you may wish to plan a strategy for when your opponent plays Y'Shaarj and take of what Corrupt cards have been played so you will know what your opponent will do when they do play Y'Shaarj.
If you're playing aggressively, then your opponent will have less of a chance to make full use of their Corrupt cards. If you suspect your opponent has Y'Shaarj in their deck, one possible strategy is to shift to an aggressive gameplan to end the game before they can make full usage of Y'Shaarj.
Example Hearthstone Corrupt Cards
Day at the Faire is a card for Silver Hand Recruit strategies. On its base form, it will summon three Silver Hand Recruits for 3 mana. When upgraded (or "Corrupted" if you will), it will summon five of them instead, effectively making it a Stand Against Darkness that costs 2 less mana. 3 mana is a fairly cheap cost, so upgrading it should be easy.
Dunk Tank in its base form will deal 4 damage for 4 mana, which is slightly underwhelming but not the end of the world if you need to cast it this way. If you upgrade it though, it will deal 2 damage to all enemy minions in addition to the 4 damage, which will equal 6 damage to a minion and 2 to all other enemy minions (or 2 damage to all enemy minions and 4 damage to the enemy hero if you target their hero).
Fleethoof Pearltusk is a minion with Corrupt. In its base form, it's a 5 mana 4/4 with Rush which is quite weak. Upgraded, it becomes an 8/8 instead.
Y'Shaarj, the Defiler is a Corrupt-synergy card, and it ultimately when you will be trying to build a Corrupt-themed deck around. When played, it will add a copy of each Corrupted card you've played this game to your hand and they will cost 0 if you play them on the same turn as Y'Shaarj himself (itself?). The neat thing here is that all cards returned will be in their upgraded form for even more madness (at the Darkmoon Faire).