SSL Problems

Hi guys, im hoping someone can help with an SSL issue I seem to be having. Ive recently switched hosts for my website www.mygadgettree.co.uk. The new host has set me up with a dedicated IP and SSL certificate.
Ive switched on SSL in admin, and edited the admi/config.php to enable https.

I notice when logging in on the site, the url will change to https but the padlock in the corner has a warning 'Contains Unauthenticated Content'. When clicking the padlock I can see that a certificate is installed but technical details shows a message 'Connection Partially Encrypted' 'Parts of the Page you are viewing were not encrypted before being transmitted over the Internet'.

Is there something ive missed? Something else that needs to be done?

Hi trejster, your social media image is coming over http while everything else is https:
http://www.mygadgettree.co.uk/image/socialMedia.jpg
-same with twitter and facebook logos.

You can check this by opening the site in Firefox. Go to a secure page > Click the place in the address bar where the padlock should be > More information > Media tab.

Also, the tab-slide-out code taken from googlecode is not encrypted. As far as I know, anything that pulls material from another site on an SSL-encrypted page will throw an error, unless the offsite page is also under SSL...which is why I'd really like to be able to turn off some elements under SSL

Our site has exact same issue. Ive done everything that I can see. www.londonundergroundpills.com

Any ideas?