Microphones send audio to other devices through microphone cables. Most high-end microphones connect using an XLR cable. These cables have a cylindrical plug at both ends and can fit into the base of the microphone as well as many mixers and audio interfaces. Occasionally one of the three wires that run through the microphone can get detached from its connecting plate. It's possible to repair these cables by resoldering the connection.
Unscrew the plugs at the end of the microphone cable. There will often be a metal section---that's the end of the plug---and a plastic casing right below it. Grip the metal part and unscrew the plastic part to detach it from the plug.
Pull off the metal casing to reveal the wires contained within the plug. There may be another plastic sheet around the wires which can be snapped off.
Examine the wires on the mic cable. There are three wires that attach to the plug (generally red, blue, and copper). Each of these wires attaches to a small metal plate. Look to see if any of the wires are detached from their plates. Pay attention to which wire connects with which plate. Write down a diagram if it helps you remember.
Press the tip of a heated soldering iron to the connection between the wires and the plates. Once the solder liquefies, detach each of the wires so the end of the plug is free from the wires.
Cut an inch of the end of the microphone cable using the wire cutters. Then unsheathe 1/2 inch of the cable revealing the three wires. Unsheathe the ends of each of the wires.
Press a wire up against its corresponding metal plates. Bring a heated soldering iron and a piece of solder up to the point of connection between the wire and plate. Touch the tip of the soldering iron to the solder so a drop of solder liquefies and contacts the wire and plate, fusing them into place. Repeat this process for the other two wires.
Snap the plastic sheathe back onto the plug, slip the metal casing over it, and re-screw the plastic bottom, closing up the plug.