What can I expect to get out of a pack?

Blizzard never officially stated the drop chances for each rarity, so this is an estimation by Reddit user ErikRK after analyzing the contents of about 11000 packs.

The probabilities for each rarity for a single card are as follows:

Common = 70%
Rare = 21,4%
Epic = 4,28%
Legendary = 1,08%
Gold Common = 1,47%
Gold Rare = 1,37%
Gold Epic = 0,308%
Gold Legendary = 0,111%

As there are 5 cards per pack, the chance of getting a legendary in a pack would be around 5%.

A reddit post from January 3rd 2016 furthermore suggests that you are guaranteed to open at least one legendary per 40 packs of the same type. For epics it is between 10 and 20 packs (the comments were undecided about the exact number).