Public certificates are used by computers to verify the identity of a system and are necessary to implement secure connections (SSL).  For instance, when you connect to Amazon.com and proceed to the checkout, a locked padlock shows up on the browser indicating that the communication between you and the website you are browsing is encrypted with Secure Sockets Layer (SSL).  Two aspects of communication and security are being implemented here.  One being the fact that all communication between you and the servers you are communicating with is encrypted, meaning that no one can intercept your communication and read it.  Second it ensures that you are communicating with the server you believe you are communicating with. 

When you buy a certificate you have to prove that you are the company that owns the domain name.  On the Internet all machines are anonymous unless they have a certificate configured.  Having a correctly configured certificate on your server will provide a more pleasant user experience when remotely connecting to the network because users will not be prompted with security warnings from the website.  Its purpose is also to ensure the user that they are connected to the right machine and prevent certain types of computer attacks.  A basic certificate will list the website on the certificate, which is verified automatically by your browser.  Advanced certificates also list the company name, address and other relevant information needed for a person to verify the company matches who they believe they are dealing with other than just by a website name.

Certificates can be used in multiple scenarios including securing email and websites.  Installing one on the web interfaces to your network will not only make remote connections more secure, but will also provide a better user experience by not forcing a user to click continue on a certificate error.