Pretty much with Andro on this one. All of your mentioned applications cover different fields of security. If you do want to know what is overlapping and what isn't, a thorough research on the subjects is good. As Andro mentioned, explaining everything here..... Would take far too long.
Also what works and what you need are different concepts. Everything works in their environment, just if it is going to be useful to you, it's a whole other story.
The devices on your list are all mostly "separate" but they would combine to provide a security environment, and there might be some overlap. Also, security isn't necessarily limited to this list either But, here are some more details:
Router with WPA encryption: Specifically, you want WPA2 encryption plus secure settings overall; limiting outside access, fully up-to-date with latest firmware and patches if there are any, etc.
"box-type" security devices: This could mean a hardware firewall, or similar. This is the one area where there may be overlap in a sense.
AV/security software: Depending on the option you pick, it may include features such as a firewall, anti-spam, etc. and can cover many security layers. (https/www.avast.com/nextgen for more info)
VPN: This would be to help protect your online privacy by masking your public IP and encrypting your connections. (https/www.avast.com/secureline-vpn is our solution, you can find more info on VPN's in general here)