Green Monolith, or Green Meanie Monolith, is a special type of Monolith added in 1.2. Unlike standard Monoliths, these do not have a fixed position, and will spawn randomly each game. To find them, you will need a KerbNet access. They will be marked as a "?", like any other Anomaly.
Unlike normal Monoliths, the Green Monolith serves a purpose in Career and Science mode. When you discover one (getting a certain distance to it either with a vessel or an EVA Kerbal), you will recieve a message in the "Worlds First Milestones" tab. The message says: "We have discovered a VERY strange monolith on (Body name). It is beaming data to the space center" and under it, there will be a purple line of text, saying the research node you have just unlocked by finding it (e.g. Experimental Motors). As you can see, it is kind of pointless to do with a full Tech tree, but it is fun nevertheless.