• Research Any Monolith
  A journey of a thousand miles begins with a single step.
  
  
• Research All Monoliths
  The depths of space hides a secret. Unlock it.
  
  
• Last Many Waves
  The onslaught is relentless. Can you hold on for 28 waves?
  
  
• Research All Monoliths with Haste
  Can you reach wisdom before 45 minutes have passed?

These are pretty much self explanatory. For the first one you need to simply research a monolith, it doesn't matter which one, and for the second achievement you need to research the four monoliths that are found in each corner of the map. The third achievement requires that you survive past the 28th wave of attacks (I'm not sure if the achievement is unlocked when the 28th wave starts or when you have succesfully beaten all the attackers of that wave). In the last achievement you need to research all monoliths before the in game time has passed 45 minutes (you can check how much time has passed in the upper left corner when the game is paused).

• Research All Monoliths (+)
  And when you gaze long into an abyss the abyss also gazes into you.
  
  
• Last Many Waves (+)
  Different Yet the Same.
  
  
• Research All Monoliths with Haste (+)
  And when you gaze long into an abyss the abyss also gazes into you.

These are pretty much the same as the previous achievements, except Research All Monoliths with Haste (+); you need to research all monoliths in less than 32 minutes.

• Point of Origin
  Last 28 waves without researching any Monoliths.
  
  
• Extract many resources
  All is used, yet all remains. (Extract more than 140 material in one round.)
  
  
• Last Many Waves Without Quarters
  When you are not born, how can you die? (Last more than 10 waves without building quarters.)

The first achievement here is basically the same as Last Many Waves except that you can't research any monoliths. The second achievement you need to extract more than 140 material in one round. I have found this achievement to not be possible in normal mode because there are only 6 areas you can extract which add up to about 126 material (each area holds 21 material). I suggest to go for this achievement in plus mode due to the one monolith that gives unlimited material from extractor areas.