Will this wireless network adapter work with this router?

[Flaxlethal]

https://uk.pcpartpicker.com/part/tp-link-wireless-network-card-tlwn781nd

My router: https://www.asus.com/uk/Networking/RTAC68U/

 

[Bumzag]

Yes.
The TP-LINK PCIe adapter supports "802.11b/g/n", and the RT-AC68U router supports 802.11a/b/g/n/ac.

You might consider this PCIe adapter http://www.newegg.com/Product/Product.aspx?Item=N82E16833704241 as it can take advantage of the 802.11ac speeds, but the two you link will work just fine.