This is the second AP on a wired LAN, which I had bought for a better coverage of my apartment. The actual router (AVM Fritz!Box 4020) which is also the first AP is located in the corner of a small room and its wireless signal is not strong enough in the other rooms. The two APs are meant to work with the same identical SSID and Password, but the coverage did not become better with the second AP. So, a few days ago, I assigned different SSIDs to the two APs in order to find out if my tablet does really connect to the TP-Link -- and found out that it doesnt. My mobile phone with an even older Android (2.3.1) does connect.

What's the working mode of TL-WA901ND?

Does it connect to the first TL-WA901ND?

If it's access point mode, does it ocnnect to Wi-Fi without secure?

There is only one TL-WA901ND, the other AP is the Fritz!Box, which is mainly the router connecting to the WAN.

Currently I use WPA-PSK/WPA2-PSK with AES. I have tried nearly everything, except WEP and the WPA/WPA2 with Radius, since I have no Radius server.

The tablet logs in without problems into the Fritz!Box which currently works in WPA/WPA2 but also in the other mode offered by the Fritz!Box WPA2 (CCMP)

You mean I should disable security? I don't want my neighbours and people on the street to be able to log into my network.
On your question about "connect to Wi-Fi without secure" -- well, the tablet connects to the AP, but can't log in because of "Authentication problems". So the connection as such does work.

I think your question for "working mode" is answered by this display on the Wireless Security admin page: Operation Mode: Access Point

As a reminder, I gave different SSIDs to the two APs, so that I can test and find out how to connect to the one from TP-LINK. When the problems are solved -- if they are solvable -- the two APs will go back to a single SSID enabling a seemless and automatic handover between the two APs.