SNMP = Simple Network Monitoring Protocol and provides a mechanism for getting and setting device parameters, counters, etc. It may/may not provide for capture of packet data and probably will not for a simple home router.
If you're looking to capture data, then you're looking for a "sniffer". There are hardware sniffers and also software packet capture tools, like "ethereal".
Note that sniffing can be more complex when devices are connected via a switch (vs. hub) as the switch will "hide" traffic, sending it only between the source and destination vs. broadcasting to all ports.
Anyway, google some of those terms and you'll find more info.