I am watching the video NTSB Animation Flight 1549 Hudson River Landing US Airways (Youtube) and at 0:36 in the video, after the bird strike, you see the following dialogue:
HOT-1: my aircraft
HOT-2: your aircraft
I was curious to know what the formalities behind this is. Could HOT-2 also claim the aircraft?
What you're quoting is the verbal portion of an important principle called Positive Exchange of Controls. This is a process of ensuring that someone (and only one) is always in control of the aircraft, and that there is no doubt about which pilot that is.
The best practice, taught to student pilots, is a three-stage handoff procedure:
Pilot A: You have the flight controls.
Pilot B: I have the flight controls.
Pilot A: You have the flight controls.
Pilot A is the Pilot Flying before the handoff, and Pilot B is the Pilot Flying after the handoff. In an emergency situation like the above, the captain (Captain Sullenberger, HOT-1, in this transcript) is responsible for the safety of the aircraft, and can assign control as they determine is best. Skipping the first line of the handoff is a reasonable way of making this happen quickly, while not giving up much in terms of clarity. The two-stage transfer also can happen whenever someone, generally more experienced or in a teaching role, wants to be in control of the aircraft. As DJClayworth points out, a CFI will commonly take the controls in this fashion if they wish to demonstrate a maneuver - no imminent danger required.
HOT-2 refers to the "hot microphone", or recorded feed from the pilots' headset mics, from the first officer. Any time there's a change in Pilot Flying (if the captain gets up to go to the bathroom, for example), the controls should be exchanged in a manner similar to what we see above. If the FO feels that they should have the controls, they can always ask. There is no standard procedure for refusing the exchange of controls, but generally, if the Pilot Flying does not reply to the request with something along the lines of "You have the controls", then the exchange is incomplete, and the role of Pilot Flying has not changed.
During the flight, there is a pilot flying (PF) who is in control of the plane, and a pilot not flying (PNF), who is monitoring communications, systems, navigation, and other things. In order to avoid confusion, or a situation where both pilots are trying to make different control inputs for the plane, it should be clear who is in control. If the PNF wants to take control of the plane, they must say "my aircraft", and the PF must acknowledge "your aircraft". The PNF now knows that they are in control, and they become the PF.
Generally, the captain would be more experienced pilot, and take control in difficult situations (like in the US Airways 1549 flight you linked to). However, if the first officer feels it necessary, he or she should be able to also ask for control of the aircraft.
