Here's how things work:
1) Your ADSL modem synchronizes with the DSLAM at like 384kbps - 10Mbps down & 128kbps - 1024kbps up.
2) Your router (which can be the same device as the ADSL modem) then makes a PPPoE connection to the ISP. The ISP's DHCP server then gives your router a WAN IP address and WAN gateway to use.
3) Your router will act as a gateway, DHCP server & DNS server. For this it would require a LAN IP address. It's DHCP server will give all your PC's on the LAN (incl WiFi) IP addresses in the private IP range, where the gateway & DNS server IP addresses on the PC's will be that of the router's LAN IP address.
If the PC/router wants to send IP packets to a destination outside of its own IP range, then it would send it to its gateway's IP address.
So for your PC's to have a functional Internet connection, you'll have to configure your PC's such that your router's IP address is their gateway IP address.
For your router to have Internet access, you'll also need a gateway IP address from the ISP.
Here's an example.
Router's WAN: IP address 196.210.123.123, Gateway IP address 41.241.123.123
Router's LAN: IP address 10.0.0.1, subnet mask 255.255.255.0, DHCP IP address range: 10.0.0.100 - 10.0.0.254
PC's IP address 10.0.100, subnet mask 255.255.255.0, Gateway IP address 10.0.0.1, DNS server IP address 10.0.0.1
Now when the PC wants to send packets to 10.0.0.101 (which is within its IP range), it doesn't send it to the gateway. If the PC wants to send packets to the Internet (which is outside its IP range) [eg. 123.123.123.123], it sends the packets to its gateway, which would be the router in this case.
In turn, if the router gets a packet that is destined for the Internet (outside its LAN IP range), it would send the packets to its defined gateway IP address.
All the network devices should have routing tables, which will tell the device how to route the packets.