Dragon Age: Origins

At level 7 and then level 14 you will be given the chance to learn specilizations (you will gain a "specialization point" at those levels and it will stay with you through subsequent levels until you spend it.) However, first you must find their trainers and convince them to teach you - or find a book that will teach you. You can "learn" specializations for classes other than your main character's. Once you unlock a specialization it stays unlocked for all the characters on your profile (and all party members on all your characters.) The specilizations and the ways to get them are listed below:

Warrior:
Berserker: Oghren can teach you if your approval is high enough or you can learn from Gorim at the Denerim Market.
Champion: At the very end of "The Urn of Sacred Ashes" questline you will learn this from Arl Eamon.
Reaver: You can learn from Kolgrim at the Wyrmling Lair (he is a member of the Cult of Andraste - the one you meet when you do the Sacreligious and/or Ceremonialist achievments)
Templar: You can learn this from Alistair if you approval is high enough - or you can buy the book from Bodahn in the Party Camp.

Rogue:
Ranger: You can buy the book from Bodahn in the Party Camp.
Bard: You can learn from Leliana if your approval is high enough or from Alimar in Orzammar.
Duelist: You can learn from Isabella at "The Pearl" in Denerim.
Assassin: You can learn from Zevran if your approval is high enough or in Alarith's Store (after "The Landsmeet" questline has started.)

Mage:
Blood Mage: You can learn this from the desire demon in the fade (in the "Arl of Redcliffe" questline.)
Arcane Warrior: You can learn this from The Presence at the Brecilian Ruins (in the "Nature of the Beast" questline.)
Shapeshifter: You can learn this from Morrigan if your approval is high enough or from Varathorn at the Dalish Camp.
Spirit Healer: You can learn this from Wynn if your approval is high enough, buy the book at the "Wonders of Thedas" shop in the Denerim Market (the same one where you can usually buy a tome for the Educated achievment) or if you have the Warden's Keep DLC you can purchase the book from a trader in the Keep after completing the questline.

A quick and somewhat dirty way to get the Elite and Veteran achievements is to start a new game in The Awakening DLC pack with a new character. The character starts at level 18 ready for a level-up after the initial fight sequence. With that level-up comes 2 specialization points that can be used to unlock the two achievements.

Yes, this is a lazy, half-assed way to do this, but it worked for me.

I've found that you don't need to seek out the people stated in the guide above to learn your specializations.

I was in the process of getting my warrior to level 20 threw the glitch in Ostagar when Duncan asks you to get the Dark Spawn Blood and Scrolls. As I leveled I found that at lvl 7 and lvl 14 you do get your points toward a specialization and your able to use the skill point right then in there.

I'm not sure if this works at any other point in the game but I know for sure it works during your stay in Ostagar if your using the glitch.

If you play through the story normally, you'll reach Level 14 when you have around one or two quests left. At this point, you'll have two specialization points. I recommend the warrior for this achievement. You'll learn Berserker from Oghren in Orzammar, and at the end of The Urn of Sacred Ashes, you'll receive Champion. You'll also receive Reaver from Kolgrim if you choose to side with him. Templar can be bought from Bodahn in the Party Camp. Once you hit level 14, simply apply two of the specializations and you'll unlock the achievement.