The phrase "dissolve the political bands" means to do away with the governmental ties that connect one group to another. This phrase is found at the beginning of the Declaration of Independence.
The first full paragraph of the Declaration of Independence states that it had become "necessary for one people to dissolve the political bands which have connected them with another." The colonies felt it was time to cut their ties to Great Britain and no longer be ruled by that nation, but instead to be "separate and equal." The writers of the Declaration went on to say that they felt the need to explain why they wanted to be separate from Great Britain, which they enumerated later in the document.