This question prompted me to wonder how it works at a major airport. Let's say four airliners all are scheduled to arrive within a few minutes of each other to a major airport, perhaps from the four cardinal directions. Things are already backed up at the airport, let's say landings are (temporarily) restricted due to an emergency landing.
Can you describe how ATC might direct the four flights and what they would do themselves?
ATC has many tools "in their belt" to keep traffic separated. When things back up, it can get crowded. Emergencies and weather regularly cause disruptions and ATC handles them according to the circumstances.
To illustrate some possibilities, I will use Seattle as an example here. You can find the airport information here.
Most aircraft coming into the area will be commercial flights on IFR flight plans, and they will follow a Standard Terminal Arrival (STAR) into Seattle. You can find this seciton towards the bottom of the AirNav page. There are different STARs depending on the direction from which the flight is arriving. Each STAR will include published holding patterns.
Listed here is the direction of arrival, the name of the STAR, and the holding fixes:
As you can see, there are 11 different holding fixes across the seven STARs. If you have one aircraft from each direction, they can each hold at the fix according to their STAR. The busiest STARs have multiple holding fixes that may be used. Aircraft may also be stacked on the same hold at different altitudes. You can read some FAA information about holds here.
Aircraft may also choose to divert. Depending on how much fuel they have and how the weather/emergency is playing out, they may not want to hold indefinitely on the GLASR arrival at LOSTT. Generally ATC will not ask for a diversion unless absolutely necessary. Among others, Portland, Vancouver, and Spokane are all fairly close to where the aircraft may be holding and could be viable alternates.
As @fooot mentions, most, if not all, holding will take place at fixes along the arrival, not within the terminal airspace. Once airplanes are being controlled by an approach facility they will resolve spacing issues with vectors. Altitude changes within this airspace can be difficult (see: NY Tracon) and the airspace requirements for holding make it quite inefficient. Once approach runs out of room to properly sequence with vectors, they'll use their landlines to call the controller at the center feeding the arrivals and tell them "We're closed -- update at 2130Z".
The center will then start the holds. Aircraft that are closest to being handed off will be assigned the holding at the closest fixes to the destination. This usually means holding at low altitudes (E.g. holding over Yardley at 8000 ft inbound to EWR). Aircraft further back will either get a hold close to them, or they'll continue the arrival and hold toward the end. These decisions are based on airspace availability and how long the anticipated holding will be. Where center has altitude blocks available, they'll stack aircraft at that fix. I've been in holding at FL240 where I have seen 10-12 airplanes below me and a few above me, all flying the same racetrack spaced 1000' vertically. When arrivals resume ATC will peel airplanes off the bottom of the hold, one at at time at intervals dependent on the spacing requested by the approach facility.
If delays become long, aircraft will begin to divert to their alternates, refuel and then wait under a ground stop or ground delay to get back in the air to their final destination.
First come first serve.
Typically ATC will direct the first plane to the lowest free altitude in the holding pattern and the next 1000 ft on top of that and so on. It may be possible that a plane comes in at a lower altitude right after a plane at a higher one and gets to skip to a lower spot (so it doesn't need to climb up).
When the runway/next segment is available ATC will clear the lowest plane to continue/land and then shift the entire stack down.
