If you are logged into a user account in Web of Science, you can save a search or set up an email alert for new papers matching the search. Once you have carried out a search, click on 'create alert' to the right of the search bar.
You can also create an alert from the search history page, using the bell icon
Clicking this will ask you to enter a title for the alert. If you wish to save the search but not to receive email alerts, uncheck the 'Send me email alerts' box. To view and configure your saved searches and alerts, click on the "Alerts" button on the top bar. Here you can change whether or not you would like to receive emails for new papers matching the search, or can re-run the search at a later date.
Finally, if you would like to share your search results with someone else, there is a "Copy query link" button under the search results, and a "link" icon in the search history. This will create a link to your current search query, which can then be shared with someone else. By default the link will be placed on the clipboard ready to paste into an email or a document.