A browser extension proxy is an add-on installed in Chrome, Firefox or Edge that sends the browser's web traffic through a remote proxy server, switched on and off from the toolbar.
Explanation
Browsers let extensions set a proxy for the whole browser, or for chosen sites, through the browser's proxy settings API. Once installed, one click routes every tab through the extension provider's server, which is more convenient than pasting each address into a web proxy page. Because the browser itself makes the connection, sites load without rewriting, so they often behave more like they do directly. The catch is the installation and the permissions. On managed Chromebooks, school accounts and work Chrome profiles, administrators usually allow only listed extensions or block proxy settings, so the install fails or the extension is disabled. On personal devices, an extension that can read and change all your data on every site is a lot of trust in one company: in 2015 researchers found the free Hola extension was reselling its users' connections through a sister service. As with a web proxy, the extension does not cover other apps, and it may not cover WebRTC either unless it also changes the browser's WebRTC policy. AnyProxy has no extension; it is used by visiting the website, which is why it works where extension installs are blocked.
In practice
Extensions are convenient but powerful: they can see the pages you visit. In 2015 researchers found that the free Hola VPN extension used its users' connections as exit points for a paid service run by the same company, and that capacity was used in an attack on a website. On managed Chromebooks the risk is reduced differently — administrators usually allow only listed extensions, so a proxy extension cannot be installed at all.
Sources: Wikipedia — Hola (VPN)
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