Ubuntu 17.10 (and then 18.04) has brought a few changes (for prior versions, see one of the other answers). There are no longer any Amazon/web search results when you search in the Dash, but the Amazon icon remains in the launcher by default. It is part of the ubuntu-web-launchers package, which can be removed with:

sudo apt-get remove ubuntu-web-launchers

At the time of this writing, Amazon appears to be the only thing in the package, so it should be fine to remove. Keep in mind that this might not be the case in the future, so it might be safer to just remove Amazon if you might want to keep other web launchers. Take a look at the list of files in the package and get rid of the Amazon ones:

sudo rm /usr/share/applications/com.canonical.launcher.amazon.desktop

sudo rm /usr/share/applications/ubuntu-amazon-default.desktop