Some historical perspective should help to shed some light on the confusion. The gist is that "backward compatibility" is the reason we still see TKIP around.
802.11-based WLANs originally came only with WEP, which was soon discovered to be too easily cracked to be sufficient for general usage in WLANs.
IEEE started working on 802.11i, an amendment to add a much more robust security framework to 802.11-based WLANs. However, in the meantime, the Wi-Fi Alliance introduced WPA with a stop-gap scheme, TKIP, as it was taking time for 802.11i to be published, and meanwhile, there were fears that the momentum of 802.11 could be broken due to poor security. (the exponential growth of 802.11 from 1999 onwards was part of the reason why the security issues suddenly became so urgent)
TKIP - temporary key integrity protocol - as part of WPA, addresses some of the main problems of WEP related to weaknesses in the use of keys. With TKIP, it was still using RC4 with relatively short keys, but at least, doing much improved key mixing to partially address the security concerns.
Backward compatibility was a major reason for the use of TKIP and other less computationally complex algorithms like MICHAEL in WPA. The issue was that the hardware in many of the WLAN devices in those days was deemed not powerful enough to perform AES encryption/decryption, but at least they could have WPA with TKIP.
Later on, 802.11i was published. WPA2 then appeared, basically incorporating the enhancements introduced in 802.11i. There are both the enterprise and the PSK modes, but definitely, WPA2-PSK with AES/CCMP is stronger than WPA2-PSK with TKIP.
For backward compatibility reasons (in case you're trying to connect a low end device that doesn't support AES), TKIP is still supported by some WiFi access points (typically, trying AES first and then falling back to TKIP; it could be called by various names like TKIP+AES). But generally, unless you really need to support that one or two old/low end devices in your network, it would be better to just go with WPA2 (PSK or enterprise) with AES/CCMP.