Verifying your site with Google Webmaster Tools gives you access to reports about your site's visibility and ensures that your site is indexed by Google on a faster / more regular basis. It isn't necessary, but it's a smart way to help your search ranking. You can find Google Webmaster Tools here:
Follow the link above and then log in (if you already have a Google account of some kind) or create a Google account (if you don't already have one). Any sort of Google account -- like a Gmail or Google Apps email address -- will allow you to log in.
Use the Add a Site button to add your site to the Webmaster account. You can come back to this page later and add even more sites to this same account if needed.
Click "Add a Site" opens a dialog box where you'll need to enter the address of your site. If you own your own domain name, enter it as www.yourdomain.com (like www.tomsterrifictravels.com; the "www" is a necessary part of the submission); if you're using a free Weebly subdomain enter it as yoursubdomain.weebly.com (like tomstravels.weebly.com; do not include "www").
Clicking continue will open the Verify Ownership section. You have to verify yourself as the owner since Google doesn't want a random person to represent your site.
It's important here that you select Alternate Methods and choose the HTML tag option.
Copy the line of HTML code that Google provides to you.
Then leave this page open and go to your Weebly account. Within our Site Creator go to Settings > SEO and paste the line of code you copied from Google into the Header box. Be sure to Save and Publish your site. If there's already code in the header, plave this code above or below it.
When you've done this, go back to Google Webmaster Tools and click the Verify button.
If you see an error, make absolutely sure you've Published your site, give it a couple of minutes, and then try again. You should now see a congratulations message.
Clicking the "Continue" link will bring you to the Site Dashboard. Now you should submit your sitemap. We automatically created a sitemap for you: it's at http://example.weebly.com/sitemap.xml or http://www.yourdomain.com/sitemap.xml in the case that you're published to your own domain name.
Either way, you just need to enter sitemap.xml into the box Google provides you when you submit.
Google Webmaster Tools has a lot of cool features to play around with after you're verified. These features can help you see how well your site is ranked You never have to use them (verification is the only significantly important thing to do here), but they're there if you want to check them out. Google provides an FAQ that covers all the features: https://support.google.com/webmasters/answer/1050724?hl=en