|A quick look online tells me the switch you bought is a simple, unmanaged, switch. It should be plug-and-play. Therefore your setup should work.|
The first thing I would check is whether or not the switch itself is malfunctioning. The simplest way to do this would be to connect two client computers and statically assign them IP's in the same subnet. Something like the following:
Subnet Mask: 255.255.255.0
Then open a command prompt window and try to ping each PC from the other. If you're not getting proper replies, chances are the switch is no good and you should take it back and have it replaced.
If the clients do reply properly to pings, try moving the uplink cable to a different LAN port on the router.
It matters not how straight the gate,
How charged with punishments the scroll,
I am the master of my fate;
I am the captain of my soul.