ok, so the packets and the ping is them trying to work out if it's your modem shitting out, or the physical lines on your network breaking somewhere. So if I wanted to see what's going on I'd set the PC in His Lordship's office and the one in the loungeroom to both ping the modem. One drops out but the other one's still talking, i know that's one specific route. Both crap out, the modem's stopped talking 'cos that's the only wiring place they have in common.

Pings are basically just tiny little "HELLOOOO" messages sent between equipment that request a hello back to check the line is working. The "packets" message you get back is the response (and there's more data in that simple response, but that's a bigger thing that needs technical divining, packet sizes and shizzle tell you if a line is degrading)