My microphone doesn't work with the webcam

No, it doesn't, you're right! The webcam app doesn't do sound; the giveaway is that there is no configuration tab for any sound device!

Other applications, such as Skype work fine with the internal mic, but if you want to check your mic is working first, start a terminal and do

then talk a bit, then hit CTRL-C.

Now do

and you should hear your voice coming back.

Clean up after the test by doing

as the file you've recorded might be quite big!