Achievement points, AP, are acquired by completing certain tasks. Gathering different weapons, armor, and magics are some of the ways to gain AP. The most common is to hit Nightmare level raids for at least 1/2 of Fair Share damage.
Why do you want AP? Because AP gives you Planet Coins so you can buy premium armor without actually paying cash into the game. The two AP grab bags that contain Planet Coins are Peasants' Tribute (costs 100AP) and King's Bounty (costs 10,000AP).
When you are just starting out you'll be struggling to gather even 100AP. To begin with, save up at least 500AP before buying any bags.
The Peasants' Tribute has several crafting ingredients plus a random chance at 2 to 5 Planet Coins. The King's Bounty has several armor prizes plus 150 Planet Coins, which are guaranteed after getting each of the unique armor prizes. HOWEVER, until you are very far along in the game you won't have a chance to reach the initial 60,000AP or so needed to start getting the 150 Planet Coins every time you by a 10K bag so focus on the random Peasants' Tribute until you have played a very long time.
In fact, the number of Planet Coins I receive at random in the Peasants' Tribute have so far come out to more per AP than I would receive with a guaranteed King's Bounty payout.