How to find out which Wi-Fi driver is installed?

$ readlink /sys/class/net/wlan0/device/driver
../../../../bus/pci/drivers/ath5k

In other words, the /sys hierarchy for the device (/sys/class/net/$interface/device) contains a symbolic link to the /sys hierarchy for the driver. There you'll also find a symbolic link to the /sys hierarchy for the module, if applicable. This applies to most devices, not just wireless interfaces.

Maybe there's a better way, but I've used lshw -class network (as root) and it gives me this output:

  *-network               
       description: Ethernet interface
       product: 82566MM Gigabit Network Connection
       vendor: Intel Corporation
       physical id: 19
       bus info: pci@0000:00:19.0
       logical name: eth0
       version: 03
       serial: 00:a0:d1:a3:87:c8
       size: 1GB/s
       capacity: 1GB/s
       width: 32 bits
       clock: 33MHz
       capabilities: pm msi bus_master cap_list ethernet physical tp 10bt 10bt-fd 100bt 100bt-fd 1000bt-fd autonegotiation
       configuration: autonegotiation=on broadcast=yes driver=e1000e driverversion=1.0.2-k2 duplex=full firmware=0.3-0 ip=192.168.2.206 latency=0 link=yes multicast=yes port=twisted pair speed=1GB/s
       resources: irq:29 memory:fc300000-fc31ffff memory:fc325000-fc325fff ioport:1840(size=32)
  *-network
       description: Wireless interface
       product: PRO/Wireless 4965 AG or AGN [Kedron] Network Connection
       vendor: Intel Corporation
       physical id: 0
       bus info: pci@0000:06:00.0
       logical name: wlan0
       version: 61
       serial: 00:1d:e0:69:28:07
       width: 64 bits
       clock: 33MHz
       capabilities: pm msi pciexpress bus_master cap_list ethernet physical wireless
       configuration: broadcast=yes driver=iwlagn ip=192.168.0.104 latency=0 multicast=yes wireless=IEEE 802.11abgn
       resources: irq:31 memory:c8000000-c8001fff
  *-network DISABLED
       description: Ethernet interface
       physical id: 3
       logical name: vboxnet0
       serial: 0a:00:27:00:00:00
       capabilities: ethernet physical
       configuration: broadcast=yes multicast=yes

You can grep for driver from that output. In my case I use lsmod | grep iwlagn, giving me:

iwlagn                 63559  0 
iwlcore                67702  1 iwlagn
mac80211              123574  2 iwlagn,iwlcore
cfg80211               87657  3 iwlagn,iwlcore,mac80211

Don't ask me what each of those mean :)

If you're interested in USB based WiFi devices then you can see the driver name by using the lsusb command:

lsusb -t