Lose a companion if his approval goes too low

You start off with Cassandra, Varric and Solas in your party - a warrior, rogue and mage. You can then unlock the following, mostly in any order you prefer (mouse-over to view spoilers):

a warrior

Blackwall, who is unlocked through The Lone Warden quest in The Hinterlands

a warrior

Iron Bull, who is unlocked through The Captain of the Chargers quest in The Storm Coast

a mage

Vivenne, who is unlocked through The Imperial Enchanter quest in Val Royeaux

a rogue

Sera, who is unlocked through A Friend of Red Jenny quest in Val Royeaux

a mage

Dorian, who is unlocked through the story quest In Hushed Whispers in Redcliffe

a rogue

Cole, who is unlocked through the quest The Forgotten Boy in Skyhold

Once you unlock Skyhold, all party members' specializations are also unlocked. But, the game currently doesn't actually add the new skill line to them until they're in your party. Just head to a random camp, grab three people, then use the Change Party option to swap in everyone else as you go. As soon as they load into the world, they'll have their shiny new skill tree ready to go.