There are many similarities between two of the covenants mentioned in Breishit - the rainbow and brit mila. Both are called 'everlasting' and are preceded by Hashem's blessing to be fruitful and multiplying. In both cases, the covenant marks a change and a new beginning: "There will never again be a flood to destroy the earth." (9:11); "Your name shall no longer be called Abram." (17:5) The language used to describe both covenants is similar with the words "covenant" and "flesh" featuring prominently.
Yet, despite the similarities, there is a fundamental difference between the two.
God makes the covenant of the rainbow with Noah, who represents all of mankind, but mankind takes no active role. "My rainbow I have placed in the cloud, and it shall be for a sign of a covenant between Myself and the earth." (9:13) In contrast, brit mila was given to the seed of Abraham and Isaac, on condition that they "shall circumcise the flesh of your foreskin." (17:11) This particular covenant is made only with their descendants, as opposed to mankind in general, and on condition that they fulfill their part of the covenant and circumcise their sons "and it shall be as the sign of a covenant between Me and between you." (17:11)