Set items can be identified by their bright green names. Roughly speaking, they drop less frequently than rare (yellow) items but more frequently than uniques (gold). You can gamble for set items, with the odds being about 1:1000 (compared to 1:2000 for uniques).
Set pieces are similar to unique items in that they have preset properties, but they gain additional bonuses when multiple pieces are worn. This achievement unlocks upon your character equipping every item within a particular set simultaneously.
When you cursor-over a set item, underneath the properties it will also display the name of the set it belongs to as well as the name of each piece within the set. In this area, if an item's name is bright green then it means you have that piece on your character, in your inventory or in your stash.
The smallest complete sets contain 3 pieces*. A certain few contain class-specific items, but most sets can be worn by any character**, even if they are clearly aimed at one class (such as Griswold's Legacy, which contains a scepter and has bonuses to Paladin defensive auras).
You do not have to find the set pieces yourself for this achievement - a player can simply drop them for you to pick up and equip, then you can return them after it pops.
A few low-level, 3-piece sets to keep an eye out for are Berserker's Arsenal, Civerb's Vestments, Clegaw's Brace, Death's Disguise, Hsaru's Defense, and Infernal Tools.
These can all be found on Normal difficulty, and having "magic find" gear equipped will increase your chances, though not by a huge amount.
*an exception is Bul-Kathos' Children which contains just 2 swords, but this is a somewhat high level set
**your character still needs to meet whatever stat and level requirements the items possess