Identifying each type of connector
Twin Plug headsets have two plugs at the end of the cable which have separate jobs. The slightly larger one is for audio and the smaller one is for the microphone on the headset. This is the most common type of connector for general aviation (GA) and commercial airplanes. Another common name for this type of connector is PJ Plugs. Things to consider with Twin Plugs are intercom jacks do not supply power to the Active Noise Reduction (ANR) feature in a headset. Therefore to power an ANR headset with twin plugs you'll need batteries in the control module of the headset.
The 6-Pin connector is the second most common type of plug. This cable has only one connection to the panel that does it all. The 6-Pin connection provides audio, microphone and power to the headset for ANR and Bluetooth functions. 6-Pin connectors are sometimes called LEMO plugs or simply panel powered plugs. These types of plugs are found in a variety of aircraft from GA to commercial, it just depends on the model and year. Most Cirrus use this type of plug. If your airplane does not have a 6-pin connector installed, kits are available to add it. This is a quick job, but one that does require a mechanic or avionics shop.
The helicopter plug is a single connection plug. It's shorter and fatter than the plugs found on a Twin Plug cable. It supplies signal to the microphone and radio. It will not power ANR or Bluetooth functions. Based on the name you can probably guess that most people flying the whirly-birds (helicopters) will need a headset with this connection. Another common name for this type of connection is the U174 plug.
The Airbus Connector is the least common type of connector that pilot's need. It's a 5 pin plug similar to the 6 pin in that it powers the headset as well as relays the audio and microphone with one connection. The reason you would need this is flying an Airbus or certain ATR aircraft. Another common name for this plug is the XLR Connector or 5-Pin Connector.