Page Description - Meta Description Tag & Snippets - classic Google Sites
Google Sites uses the page description for various pieces of metadata, which includes the Open Graph Protocol (for Facebook Likes), Google-recommended schema.org microdata (for Google+ shares and +1's) and the standard description meta tag (search engines often use the meta description tag for search engine result pages snippets).
Sign in to your classic Google Site (this is not available in new Google Sites)
Navigate to the page you want to add the description to
Use the More button then Page settings
Add your description to the Page description box in the Page Settings dialogue box and use the Save button.
That's it, you're done.
Do not try to fill the page description with meaningless key words, make it a true description/summary of the page's content. Search engines can easily detect spammy content so they can choose to ignore the page description. Search engines may even decide to display a snippet that differs from your page description if they feel your page provides better text elsewhere on the page that describes its overall aim.
You can find more advice, from Google, about page descriptions in the video on this page.
Further Search Engine Optimisation Advice
For more SEO advice see Google Sites SEO Guide.