How do airports detect drones?
The number of drone sightings near airports is increasing (apart from last year), and frequently this results in temporary interruption of operations at airports, with enormous economic consequences. Most famous example is the December 2018 drone incident at Gatwick  which affected the airport operations for more than one day.

source: EASA Drone Incident Management at Aerodromes(pdf)
I live in Germany, where there have been over 500 sightings near airports since 2015, including several high-impact cases that resulted in the interruption of air traffic at major airports such as Frankfurt airport. Last year, the German Air Traffic Control (DFS) tested a number of drone detection systems  (press release, video link, article in German)  at Frankfurt and Munich airport as part of a market evaluation. Over 600 drone flights were performed, at distances up to 10 NM from the airport. I was involved in the design and execution of those tests and I can share a few observations.
At most airports in the world, drone detection still primarily relies on humans that happen to see a drone fly and inform the police, airport security or control tower by phone or radio.  There are a number of severe limitations to that approach, for example:

Drones are typically small and difficult to see at distances of more than 100 meters. At the scale of a major airport, it is plausible that drones can fly in safety critical areas for a prolonged time without being seen.
Position reporting to those that need to act upon the information (air traffic control, police, security staff) is delayed (phone call, relay of info) and inaccurate (&quot;near the terminal&quot;, while the terminal building stretches over more than a kilometre).
Independent verification of the reports is difficult. This opens up the possibility to falsely report drones in an attempt to shutdown the airport.

To address these limitations, drone detection systems are under development, and some airports have deployed them with varying levels of success.
Ideally, a Drone Detection System (DDS) meets these high level requirements:

it detects all drones in the vicinity of the airport
it locates these drone(s) accurately
it does so without delay
it has a very low false alert rate (preferably no false alarms are raised at all)
it locates the position of the remote control (not needed for safety of air traffic, but helps the police in finding the drone pilot)
it does not interfere with aviation communication, navigation and surveillance systems.
preferably the system does not need any supervision; having staff 24/7 dedicated to drone detection is very expensive.

There are many drone detection systems available in the market, at various levels of maturity and some more suitable for airport use than others.  Most, if not all drone detection systems, make use of various sensor types.
There are four categories of drone detection sensors:

Optical / infrared cameras
Radars
Radio detection systems
Acoustic detection systems

Optical / Infrared cameras
One way to detect drones is by cameras. But at the size of a major airport, you are almost always far away from the drone. This means that the camera must be able to zoom, zoom a lot. Some of the best cameras are able to show a clear picture of a 40 cm size drone at 2 kilometres, and a recognisable dot (distinguishable from birds) at several kilometres more. The problem is that with such zoom levels you have a very narrow field of view and so you are never going to detect a drone that you didn't know about before. Therefore cameras are not a good choice as primary sensor, they need to be pointed in the right direction by another sensor.
Once they are pointed at the object of interest, image recognition is often able to classify the object as drone or something else. Some camera systems do a great job optically tracking moving drones, others need manual input.
There are systems that use the camera subsystem for locating the drone, by combining the view angle of several cameras to calculate where the lines-of-sight cross. Laser range finders can also be used in combination with cameras to determine the distance of objects from the camera. Other systems just use the camera as means of classification of an unknown object, not for determining the position.
Having a combination of Infrared (IR) and visible light is beneficial. For one, because at night there is not enough visible light to see the drone, but also the contrast of the drone against the background is often much better in IR than in visible light.
Radars
Radars are an obvious choice of detecting flying objects near the airport. Unfortunately, air traffic control radars and surface movement radars are not able to detect objects of the size of a drone.
There are dedicated drone detection radars on the market, some with a background in military, border control or bird tracking, that are specifically tuned for drones. The maximum distance at which these radars detect a drone of course depends on the size of the drone, but also its shape and the manoeuvring it is doing. There are significant performance differences between radars, and even a good radar may perform badly if it is not properly tuned or sited.
Since these radars have to be very sensitive to pick up a radio reflection from a drone, they see almost everything: planes, cars, cyclists, birds, wildlife, trees moving in the wind etc. One of the biggest challenges for drone detection radars is to distinguish drones from other objects.
False alarms need to be balance with missed detections, which is a delicate process. In the end, a radar will occasionally classify something as drone while it is actually something else.
Radars transmit radio waves that potentially interfere with CNS systems at the airport. Most systems I have seen were operating at frequencies far away from aviation systems, and those that were operating at similar frequencies did not cause interference. (After doing a paper analysis first, the systems were switched on to see if there was interference. This was done at night when air traffic levels were low, and with all relevant people made aware before the system was powered up)
Radio detection
Radio detection systems work by detecting the radio signals transmitted by the drone or their remote control. These systems contain a library of signal properties and they recognise which signals belong to drones. Most systems I have seen cover vast frequency ranges.
The location of the drone (or its remote control) can be found by triangulation or multilateration. Some systems can decode data packages from specific drone models and read the GPS location directly from the drone and the remote control.
Since there are many radio signals transmitted at the airport, the background radio noise is quite high. Also radio signals reflect of large buildings such as hangars and terminals. This has an effect on the performance of radio detection systems. Some systems are severely impacted by the airport environment while other systems are less affected.
Of course these sensors only work if the drone transmits a signal, does so on a supported frequency, and with a protocol known in to the signal library. None of the systems I've seen was able to detect all drones in the test.
Acoustic
Finally there are acoustic detection and location systems, operating in the acoustic frequencies band.
These are passive systems that rely on detecting the noise characteristics of drones (specifically their propellers). They have a very limited range (up to ± 300 m) and are  susceptible to background noise, which degrades their performance.
Due to these characteristics the use of acoustic detection systems in airport environments is very limited. If there is any use at all, they could perhaps be used as a coverage gap-filler, for example between hangars.

A good drone detection system will make use of various sensors types and integrate their outputs using a multi-sensor data fusion and tracking system to a single, reliable situation picture. In every class of sensors there are some really promising products, but a  strong integration of a range of good sensors is hard to find. This market is developing quickly and I expect that the situation will be very different in a few years.