The suppress contacts button allows you to block further outreach to selected contacts.
Another way to suppress contacts is to go to settings > suppression and to add specific email addresses or entire domains to the suppression list. In case of an entire domain, all the actual and future contacts with an email belonging to that domain will be excluded from future sendings.
Please note that only account admins can use the suppression-list feature.
A detailed article for this process can be found here