Every time I try to install an app from the market using my laptop web browser, I have to select the device to download to. The options are "Galaxy Nexus" and "phone". Why are there two options? Aren't they the same thing?!
To answer your question regarding - "aren't they the same thing", they might or might not be. Google keeps track of all devices that you register/log into using your google email acct. If you had a previous google/android phone, it might be showing your previous phone in the list in the Market also. If you currently only own the Galaxy Nexus, there is a way to hide the "phone" listing in the Market. To do so, log into the Market and then select - My Library, Apps, My Orders, Settings in that order.You should now see all devices listed that you have owned/logged into using your google email acct. To remove the "phone" listing, click on the Edit button to the right of that option and should now see two options under "Visibility" - Hidden in menus & Show in menus. Select the Hidden in menus option and then select Update to the right again. If done correctly, now whenever you try installing an app, you should only see the Galaxy Nexus option listed. Hope this helps.
PS: As far as I know, there is currently no way to permanently delete a device from the Market listings.