The resistance logo is a red double helix. The helix represents DNA and the human nature of the resistance. The color red represents all the blood that his been shed in fighting the machines.
The patch is in the shape of a shield showing that we will defend ourselves. The yellow sword emphasizes the offensive nature of the resistance that will ultimately take the fight to the machines taking down Skynet, Genisys, Legion or whatever other form of artificial intelligence that presents itself as a threat with different goals than humanity.
While this is the official logo, many use a variation. In graffiti form, the red double helix is the primary method of tagging and marking friendly locations.
During battlefield operations, a red armband is worn allowing for a better far recognition signal by other members of the resistance. Just like the representation of the red double helix, the red armband represents all the blood that is and will be shed in taking back the earth for the human race.