Sorry I should of stated to stop the ping after 4 or 5 times.
Well the good news is that we know your router is handing out ip addresses & because you can surf it's also correctly setting your default gateway & DNS.
Now lets take a look at your wireless card 'system preference > network'. Once your in the network gui if still connected via your Ethernet it should be green? if on the other hand you've disconnected then it'll be red. What status is your wireless card? btw I'm not aware of your level regarding 'networking' so if I'm teaching you to suck eggs I apologise
Ethier way we now want your wireless card to state 'Connected' so turn on your 'Airport' & probably best to disconnect your Ethernet first. On a side note what other 'services' do you have listed to the left with 'Airport' & 'Ethernet'
Once your 'Airport' states 'Connected' go into the 'Advance' tab & tell me which 'Preferred Netwoks' do you see regardless remove all of them other than your home. Then move to the next tab 'TCP-IP - what do you see there? You should see IPv4 Address then an address along with your 'subnet mask' & 'router' address?
Then move to the next tab 'DNS' what do you see?