I am trying to use an ethernet cable cord and a ethernet network splitter connector coupler plug for my desktop and laptop. When I plug them both into the coupler my internet gets disconnect. Can I do that or is there something else I need to do.
Thanks

You want to use a "network switch" typically. The most basic ones are quite inexpensive. It allows you to connect a source cable and as many device cables as you wish.

Amazon.com: NEW TP-Link 8-Ports 10/100 Fast Ethernet Internet Network Switch: Electronics

8 Ports 10/100 Mbps Fast Ethernet Network Switch/Hub - N01 - Buy.com

D-Link Systems DES-1105 - Tech for Less


You could opt for a little more power by using a Router:
Linksys BEFSR41 Cable/DSL Router - 10/100 Mbps 4-Port (Refurbished) at TigerDirect.com