Hearthstone - Golden Cards
Golden cards are special, rarer versions of cards. Each card exists in both a regular and a golden form.
The difference between a card's normal version and its golden version is purely aesthetic - all golden cards feature a golden frame, and each has a unique animation.
There is no difference in gameplay or behaviour on the battlefield, and golden cards do not allow you to circumvent the 2-per-deck (1-per-deck for legendaries) card restriction.
Golden cards are commonly compared with "foil" cards of other Trading Card Games, and are far harder and more costly to obtain than their regular versions. Golden cards are intended to allow players to show off their card-collecting achievements.
While they may contain no tactical advantage than regular cards, the main and most important difference is that they will grant more dust during disenchanting; which varies by rarity
Hearthstone - Crafting
As @Sconibulus mentioned, you have a chance to obtaining a Gold Coin if your deck consists entirely of gold cards.
Hearthstone - The Coin
If a player's deck is composed entirely of golden cards, they will
be granted a golden version of The Coin when going second.
Another cool aspect, which is purely cosmetic, is that if you play a gold card that replaces or upgrades your Hero Power, you will obtain a gold hero power
Replacing Hero Powers (cited from Gold Card link)
- Cards which replace a Hero Power with another one (such as Lord
Jaraxxus, Sir Finley Mrrgglton or Shadowform) will only grant golden
Hero Powers if the card is golden. For example, playing a regular Sir
Finley Mrrgglton with a golden Hero Power will replace it with a
regular Hero Power; playing a regular Shadowform after playing a
golden Shadowform will replace the golden Mind Spike with a regular
- The one exception is Justicar Trueheart, which will produce a golden Hero Power if either the card or the Hero Power is golden.
This may be because the card conceptually "upgrades" the Hero Power.