Click on the Ubuntu button, search for "Privacy" and then turn off "Include online results" :

You can also remove it entirely by clicking on this button, then the orange "Available on the Ubuntu Software Center" button and selecting uninstall:

or via the command line

sudo apt-get remove unity-lens-shopping

You then need to log out and back in.

Another option would be to move the lens entirely out of the home dash and make it a standalone lens:

• How can I move Amazon search results into a dedicated shopping lens and out of the home lens?
• I have permanent connections to Canonical servers, what are they for and how can I turn them off?

If you don't want online shopping results in your dash but there are times when you want to search the Internet from your queries (for example, while using the Askubuntu scope to the help lens), you may not wish to disable the "Include online search results" item. In that case you may wish to remove particular sources.

For Ubuntu 13.04 and Earlier

Home Lens

For the new dash home lens (default) queries that are forwarded through Canonical to Amazon uninstall unity-lens-shopping.

Either click on the bag icons above and click the Remove button on the right in the Software Center window that appears, or use an alternative of the following command line after bringing up a terminal with Alt+Ctrl+T.

sudo apt-get remove unity-lens-shopping

Or, launch Ubuntu Software Center, click Installed, expand Themes & Tweak, and scroll to the its end. You'll see Shopping lens for unity there (along with a lot of complaints).

Video Lens

For the already existing video lens lookup you can choose the source of searches by using the "Filter Results" item on the top right, and then selecting the sources you want by using the Sources drop down. Alternatively you can uninstall unity-scope-video-remote.

Either click on the bag icons above and click the Remove button on the right in the Software Center window that appears, or use the following command line after bringing up a terminal with Alt+Ctrl+T.

sudo apt-get remove unity-scope-video-remote

Music Lens

As in existing releases, online shopping for music in 12.10 can also go to the Ubuntu One Music Store. To remove that feature click on the bag button for unity-scope-musicstores and click on the Remove button on the right in the Ubuntu Software Center window that appears.

or enter the following command in a terminal:

sudo apt-get remove unity-scope-musicstores

For Ubuntu 13.10 and Later

(From http://www.webupd8.org/2013/10/how-to-disable-amazon-shopping.html)

Go to the Dash by Pressing the Super-A key combination. For PC hardware the Super key may be the one that has the somewhat familiar computer company logo on it.

Click on "Filter" on the upper right of the dash, and then on "Type" right below. Then click on Dash Plugins to display them.

By clicking on the plugins of interest on the left you can get a summary of each plugin, including the Amazon one. You can disable or enable them individually with the button that appears below the summary.

For enhanced privacy the source for this answer suggests the following list for consideration.

"Amazon, Ebay, Music Store, Popular Tracks Online, Skimlinks, Ubuntu One Music Search and Ubuntu Shop."

I would add the Ubuntu One Music Store, and Remote Videos as well.

There may be other plugins you don't care for as well, particularly since there are now a great number of scopes and many may forward your interests to sites you haven't vetted.

unity-lens-applications must be installed for this approach to work.

Usually when you bring up the dash you start with the Home lens. You can adjust what is searched by altering the Filter settings on the upper right of the dash before you starting type your search term.

This has changed from previous versions where all you had to do was

sudo apt-get remove unity-lens-shopping

However that does not work with 13.10 (and presumably later versions when they are released).

Ubuntu 13.10 does not have the shopping lens any more but now has Smart Scopes instead.

To disable a dash plug-in, From the dash select filter results then "Dash plugins"

To disable a plugin, click on it and then click the "Disable" button. You can re-enable it later if you want.

You can also disable all these plugins from the command line with

gsettings set com.canonical.Unity.Lenses disabled-scopes "['more_suggestions-amazon.scope', 'more_suggestions-u1ms.scope', 'more_suggestions-populartracks.scope', 'music-musicstore.scope', 'more_suggestions-ebay.scope', 'more_suggestions-ubuntushop.scope', 'more_suggestions-skimlinks.scope']"

_{Source here:How To Disable Amazon / Shopping Suggestions And Other Unity Scopes In Ubuntu 13.10}

This stops the scope running but does not remove it from the dash

If you want to do this first unlock it from the launcher by right clicking it and selecting "unlock from launcher".

You will still be able to search for it in the dash but you can remove it from there too.

You need to run nautilus as root to do this See my answer to this question Why is gksu no longer installed by default in 13.04? for instructions on installing gksu or a workaround to run GUI apps as root.

You can now run the file manager with gksudo nautilus

Naviigate to /usr/share/applications find the file with the amazon logo and delete it.

Finally reboot and its gone for good.

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

With Privacy Indicator you can quickly enable or disable online results:

In addition to the other answers, you can also redirect the queries to another location by changing the URL it uses for the product search:

sudo -s
echo 'OFFERS_URI="https://localhost:0/"' >> /etc/environment

And then restart your session.

I am not sure about the implications of this solutions but after removing shopping lens from Ubuntu 13.10 and higher (presumably)

sudo apt-get remove -y unity-lens-shopping

and disabling the scopes

gsettings set com.canonical.Unity.Lenses disabled-scopes "['more_suggestions-amazon.scope', 'more_suggestions-u1ms.scope', 'more_suggestions-populartracks.scope', 'music-musicstore.scope', 'more_suggestions-ebay.scope', 'more_suggestions-ubuntushop.scope', 'more_suggestions-skimlinks.scope']"

as well as disabling remote search (this way your searches won't be sent to some remote location)

gsettings set com.canonical.Unity.Lenses remote-content-search none

you can also manually remove the link from dash ( please comment if this is the correct way or there is a better way of doing this )

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

In 14.04 LTS, I was able to uninstall the Amazon application by:

1. opening Dash
2. searching for "Amazon"
3. right clicking the Amazon application icon
4. selecting "Delete" from the application's description

IF you want to have a Ubuntu LiveCD without Amazon stuff there is a remastering script: http://www.helplinux.ru/wiki/en:kb:make-ubuntu-safe

This command lists all the scope files that can be disabled:

find /usr/share/unity/scopes/ -name \*.scope -printf "'%P',"|sed -es':/:-:g' -e's/,$//'

There are some scope files missing in the above command like 'code-manpages.scope', 'grapics-openclipart.scope', 'video-remote.scope', 'video-local.scope', 'help-yelp.scope', 'reference-zotero.scope'