Browser and server security are important due to vulnerabilities that can be exploited by hackers. Browser security involves protecting against attacks like cross-site scripting and sensitive data exposure, while sandboxing and isolating processes. Server security requires securing the operating system, software, network, and data, as well as securely configuring, patching, and monitoring the server. Best practices for both include auto-updating, HTTPS, and blocking plugins/popups for browsers and least privilege access and logging for servers.