Hoardy-Web passively captures and collects dumps of HTTP requests and responses as you browse the web, and then archives them using one or more of the following methods:

by saving them into your browser’s local storage, which is the default;
by submitting them to your own private archiving server via HTTP, like
- the advanced archival+replay server hoardy-web serve (also on GitHub) or
- the simple archiving-only server hoardy-web-sas (also on GitHub);
by generating fake-Downloads containing bundles of those dumps, making your browser simply save them to your Downloads directory.

Hoardy-Web produces dumps in a very simple, yet efficient, WRR file format (also on GitHub). Moreover, Hoardy-Web implements:

UI indicators that help in ensuring good and complete website captures,
optional post-capture machinery that helps in archiving only select subsets of captured data;
archival+replay integration when used in combination with an advanced archiving+replay server like hoardy-web serve (also on GitHub);
capture of DOM snapshots;
inspection of captured HTTP traffic, similar to browser’s own Network Monitor (except browser-wise, i.e., it can capture HTTP requests even when a page generates them when its tab/window closes; which a surprising number of websites does to prevent their web traffic from being inspected with Network Monitor);
some generally useful web browsing features (like per-tab Work offline mode).

In other words, this extension implements an in-browser half of your own personal private passive Wayback Machine that archives everything you see, including HTTP POST requests and responses (e.g. answer pages of web search engines), as well as most other HTTP-level data (AJAX, JSON RPC, etc). For more information see project’s documentation (also on GitHub) and extension’s Help page (also on GitHub) (also distributed with the extension itself, available via the “Help” button from its popup UI), especially the “Frequently Asked Questions” section there (also on GitHub). Also, note that:

Hoardy-Web DOES NOT send any of your captured web browsing data anywhere, unless you explicitly configure it to do so.
Hoardy-Web DOES NOT send any telemetry anywhere.
Both of the above statements will apply to all future versions of Hoardy-Web.

Hoardy-Web was previously known as “Personal Private Passive Web Archive” aka pWebArc.