You can think of an airport's capacity as the supply of available landing or takeoff slots. A slot can only be used for a landing or takeoff, so an airport must to favor arrivals or departures. Important technical note: it's not the airport authority - or airlines, for that matter - that set an airport's capacity, but the Federal Aviation Administration (FAA), though the former parties may exercise some influence. The FAA is generally inclined to favor arrivals - once airborne, a plane is eventually going to land somewhere. Understandably, the FAA is highly motivated to ensure a runway slot is available. With the interdependency between arrival and departure capacity briefly treated, we'll make the following over-generalization for the sake of this explainer: an airport's arrival capacity is set first and the remaining non-directional capacity is left for departures.
Arrival capacity is typically expressed as an hourly rate: the number of arriving aircraft the airport can handle per hour. So what variables affect how many arriving flights the airport can handle? And how can a given variable increase or decrease an airport's capacity? We think it's a perfect problem for deep learning to solve! Unfortunately, we're still building those algorithms.
While those algorithms are under construction, we can still offer some insight. Let's consider wind direction first, as this will determine the direction that traffic flows at the airport. The FAA will select a runway configuration that allows aircraft to takeoff and land into the wind (this uses less runway). Runways are assigned a number that is associated with their heading, e.g. a runway aligned due east will be number 9 (090 degrees). The vast majority of runways can be used in both directions, so the pavement associated with the hypothetical runway 9 above can oppositely be used as runway 27 (270 degrees). Like a runway direction can be expressed in degrees, so too can a wind's direction; to confuse matters, while a runway's degree reflects the direction its headed, a wind direction's degree indicates the angle it's from. This apparent contradiction actually aligns runway and wind direction, so that a 270 degree wind angle allows aircraft to takeoff and land using runway 27 (again - into the wind). One important exception: when winds are underneath 5 knots (or approximately 6 mph), the FAA can decouple runway selection from wind direction.
Airport planners optimize for the prevailing wind direction when designing runway complexes, however this means an abnormal wind direction can force a sub-optimal runway configuration. Denver Airport is a good example: they have 4 north-south runways to support the prevailing southerly wind, however only 2 east-west runways. On the occasion that an easterly or westerly wind materializes in Denver, airport capacity is automatically cut in half. For our curious readers, FAA airport diagrams can be found here. So if wind direction determines which runways are available, what determines how many slots are available on each runway? The short answer is visibility.
Visibility - coupled with ceiling, i.e. the lowest layer of cloud that mostly or entirely covers the sky - dictates the rules that aircraft must operate under. If visibility is less than 3 miles and/or ceilings are less than 1,000', then aircraft can no longer maintain visual separation from other aircraft. Absent aircrafts' ability to visually separate from each other, the FAA is required to increase the spacing between arrivals, thus reducing the supply of slots. Snow is the last weather factor that warrants mention, as snow removal efforts require that a runway be closed.
So far, we've concerned ourselves with weather conditions within a few miles of the airport. We could write enough about the impacts of surrounding weather (i.e. within 40 miles from the airport) to fill its own explainer, but for now we'll leave it at this: vicinity thunderstorms can close avenues to or from the airport, thus indirectly reducing an airport's capacity.
We've pretty well covered the impacts of weather - what does that leave? Runways require a fuller rehabilitation periodically (on the order of years), which closes the runway for months at a time. While a bit buried, your best bet for identifying any capacity reductions resulting from a runway rehab will likely be the Construction Impacts Report published quarterly by the FAA. Unfortunately, there's also construction (or more aptly, repair) of the unscheduled variety. Equipment that enables non-visual operations can fail or get damaged, which likely reduces a runways usability; pavement can also fail (yes, as in pothole), which is liable to close the runway.
The last factor we'll mention is the mix of aircraft types. Aircraft generate turbulence in their wake and the intensity of the turbulence is proportional with the size of the aircraft. Larger aircraft (think those with 4 engines or the largest 2-engine aircraft) generate enough wake turbulence that air traffic controllers generally are unable to use the following slot. The influence of aircraft types on airport capacity is less noticeable, though can make a difference on the margins.