OK lets all take stock and use some logic here and I respectfully disagree with a few of the comments generally being banded about ! And i overall am unhappy about the general forum bashing as well as people may find a hint useful even though 'you' have not.
WPA-PSK / WPA2-PSK does not require an 'authentication server' per se as @leamanc has suggested.
Pre-Shared Key (PSK) Mode
The PSK mode is designed for home or small network use where an 802.1x authentication server isn't required. WPA-PSK works by regularly changing the automatic encryption keys authenticated between computers, your server or router and other devices that connect to it. PSK dramatically improves protection over WEP, as the encryption keys change very quickly, thus preventing intruders from gathering data to break encryption and into your network.
Now lets get to Encryption.....
WPA tends to use TKIP and WPA2 uses AES encryption.
Both these methods Apple call WPA Personal or WPA2 Personal.
AES is preferred as it is more secure and recent exploits of TKIP have led to it being slowly phased out just like WEP which should not be used at all.
However WPA works with most older hardware where as WPA2 will require more modern hardware (6 year old ish)
Overall with modern Wi-FI hardware using 'n' protocol you need to ONLY use WPA2 with AES encryption to get the best speeds/throughput. But that is another story.
I do not really want to go into WPA2 Enterprise, RADIUS, 802.1X etc as this is just going to make it even more complex for people to understand and irrelevant for Home users as this used when connecting to Business/Enterprise networks.

How To Crack Wpa2 Wifi

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Crack Wifi Password Wpa2

Now we are disconnecting a user from wifi to capture a handshake. Type this command: aireplay-ng -0 3 -a A1:B2:C3:D4:E5:F6 -c wifi-user bssid wlan0. Here, -a is to define wifi bssid(mac address) and -c is for a user who is connected to the wifi. And -0 stands for deauth-attack(deauth-packets value is 3).