Generally speaking, it can be said that Kurt Godel showed via his Incompleteness Theorems that any formal system based on axioms that are sufficient for arithmetic can be manipulated in such a way as to make the following like statement from entirely within the logical mechanisms of the formal system:
This formal system will never determine that this very statement is true.
The above statement (call it G for Godel) will be shown to prove that the formal system (or any formal system) is necessarily incomplete (i.e. it can not state all truths from within the mechanisms of the formal system). This is the case because G is in fact true but the formal system will not be able to say that it is.
Now, any formal system displays its limits (its incompleteness) when it is directed to determine whether or not G is true, which is a proper directive for a formal system. This is because if the formal system responds that G is true then G is, in fact, false (i.e. what G asserts would become false because the formal system would have now made a determination that G is true, which is contrary to what G asserts).
Therefore, if G were false then the formal system would have made an incorrect and contradictory determination by determining that G is true. It is because of this situation that the formal system will, indeed, never be able to make a determination as to whether G is true. So it is then the case that G is in fact true but the formal system will never be able to say that it is, leading to the formal system’s inherent incompleteness.
Gdel’s genius was discovering how any requisite formal system could be manipulated through proper logical mechanisms to make a statement G (see Godel Numbering). It is this discovery that is often said to rank among the great intellectual achievements in human history. How Godel was able to manipulate any formal system to create G is highly technical and is where most laypersons will depart their inquiry.
The consequences of Godel’s Incompleteness Theorem are not universally agreed upon and are a long way from being completely understood. In example of a consequence that is largely agreed upon is that Incompleteness makes utter mathematical certitude beyond reach, a consequence that completely undermined the work and hopes of many leading mathematicians during the first half of the 20th century. In example of a consequence that is not largely agreed upon is the idea that Incompleteness proves that the human mind cannot be shown to be a computer like machine due to the fact that the human mind will always be able to find truths that a computer like machine will not be able to find, as computer like machines are based upon formal axiomatic systems.