Two Eyes

To be safe from capture, a group of stones must in general have two eyes ^[1]. This is the primary way to make groups alive (the other, usually less desirable way, is seki).

The basic rules of go state that when a group's last liberty is removed, the group is captured. It follows that if a group has at least two separate ^[2] liberties which cannot be removed, the group cannot be killed and is alive. All subtleties aside, such an unremovable liberty is what we call an eye.

Two Eyes  

Look at the circled points. A white stone placed on either of them would be disconnected from the other white stones and would have no liberties. Both black chains would still have the other circled point as a liberty. Such a move by White would thus be illegal. Neither of the two circled points can be occupied by White without the other. In conclusion, the circled points are two unremovable liberties of the black group, or two eyes, and it is impossible to remove the black group from the board (without Black filling one eye).

• Check out the Smallest Group with Two Eyes.
• There is a formal definition of two-eye-formation.
• Note that sometimes a group may have a single eye with more than one point in it, but that does not neccessarily make the group alive: separate eyes are required.
• Often, a liberty that appears to be an eye may actually be removable by the capture of a chain of stones; see False Eye.
• Read more in Eyes Collection and Life and death
• Check out the Two Squares patterns

[1] A group is generally considered to be alive if it is able to make two eyes in response to any threat.

[2] If a group has two liberties which are adjacent to each other forming a single eye, and no other liberties, then it's dead. However, it's possible for a group to live with two adjacent liberties, if they're also adjacent to another group, forming a seki.