You will need a variety of graphics in different dimensions to have for the website, social media, letterhead, and business cards.
Tip: Consider having a version of your logo with the location. So people know where you are in the world.
1. Square Logo 800x800
(Facebook, Twitter, Pinterest, Google+, LinkedIn, Instagram, YouTube Channel). If the designer makes the image 800x800, the logo can then be resized for the requirements for all social media. The sizes are: Facebook 180x180, Twitter 400x400, Pinterest 600x600, Google+ 250x250, LinkedIn 200x200, Instagram 612x612, YouTube 800x800.
2. Icon logo
152x152 (Shortcut for Iphone and Android)
3. Favicon 16x16
(This is the little icon that you see in browsers) It is possible to use the icon logo above as your favicon. As long as it looks good small.
4. Background photo or image for Social Media
If you start with a size of 2560x1224, it can be resized to cover all the social media. You may want to use the same image for branding purposes. Here are the specific dimensions Facebook 851x315, Twitter 1500x500, Google+ 2120x1192, LinkedIn 200x200, Instagram 646x220, YouTube 2560x1224
5. Logo for website
(which could be the square one above or designed into a banner image). The logo dimensions depends on your template layout.
Graphic Design Cheat Sheet from Constant Contact