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 Silence keyword.
Removes all card text and enchantments.
Silence is a mechanic that will remove all effects and enchantments from a minion. After a minion is Silenced, any effects on it will no longer apply, although it can still have other enchantments applied to it later. Returning a Silenced minion to your hand, or resummoning it through other means will "refresh" the minion back to its base form.
Silence is a sparsely used keyword due to relatively dangerous design, and Silence effects are usually (although not always) specifically dedicated to a specific type of target.
Silence is often associated with Priest, although through Neutral cards, any class can use Silence effects.
Keywords that Work Great with Silence
Silence can get rid of any negative Deathrattle effects on your minions, although such times are quite niche.
Playing Against Silence in Hearthstone
Silence cards are rare and are typically not played. As such, Silence is rarely a problem.
That said, if Silence is prominent in the meta, you may wish to avoid playing cards with large Deathrattles or putting several enchantments on minions until your opponent has expended their Silences. You can also spread your enchantments around multiple minions, as even when Silence is part of the meta, it's rarely possible for your opponent to Silence more than one of your minions at a time.
If a card of yours is Silenced, you can return it to your hand or "refresh" it with cards like Wretched Reclaimer to summon a new copy, "un-silenced".
Example Hearthstone Silence Cards
Spellbreaker is a simple way to Silence any minion, and has since been moved to the Hall of Fame.
Shieldbreaker is a conditional Silence effect that can only Silence Taunt minions. Aggro historically would use a Silence exclusively for Taunts, although the card had not seen play.
Purify is a card that can only Silence one of your own minions. The card was extremely controversial upon reveal and was regarded as one of the worst cards in the game. Despite this, it has actually seen legitimate play in a real competitive deck in Silence Priest.
Plague of Death is a unique Silence card that Silences absolutely everything on the board and kills it, nullifying all Deathrattles and other effects that go with it, giving you a clean slate to work with.