This achievement requires that you upgrade a MEC to level 3 (its maximum). The first thing you will need is an Augmented Solder to put in your MEC armor. You can make one in the Engineering --> Augment Soldiers menu (once a Cybernetics Lab has been built). The soldier you pick will lose their class but gain a passive bonus based on their original class (
http://xcom.wikia.com/wiki/MEC_Trooper for details).
Note: MECs can't use items but gain some powerful weapons and abilities - I personally took one to every battle. Any more may be too limiting due to Meld costs and lack of items, but they can be real killing machines.
OK, you have an augmented soldier - now to get him/her a MEC suit! To do this you will need to complete specific research to enable each tier:
Level 1: Meld Recombination
Level 2: UFO Power Source
Level 3: Titan Armor
Once the appropriate research is complete, you will also have to build/upgrade the MEC in the Engineering --> Build/Upgrade MEC menu. Prices are as follows (estimated, money costs can differ based on workshops - MELD costs do not change):
Initial level 1 build: $25, 40 Meld
Level 2 upgrade: $30, 60 Meld (requires 10 Engineers on staff)
Level 3 upgrade: $110, 100 Meld (requires 25 Engineers [thanks for the correction horstwayne!])
Once you have a MEC upgraded to level 3 this achievement will pop!