Android

Application stores

APKMirror - Trusted APK downloading website.

Aurora Store - Open-source alternative to Google Play Store focused on providing privacy and modern design.

F-Droid - Decentralized store for downloading applications.
        Neo Store | Droid-ify - Alternative stores to the official application with Material You design

Browsers

Firefox Nightly | Fennec - Nightly and Stable versions of Firefox.
Here's a Guide to add more extensions to Firefox Nightly.

IceRaven - Fork of Firefox which provides additional features, removes Mozilla Telemetry and adds more extensions to the current list, allowing you to make your own too.

Kiwi Browser - Chromium-based Web Browser that supports extensions.

Privacy

PowerTunnel - Powerful and extensible DNS server with anti-censorship functionalities for Android.

AdAway - Open source mobile ad-blocker with a focus on privacy.
Benefits from having Root access.

File Managers

MiXplorer - Fully-featured Android file manager that supports multiple cloud storages and server protocols.

Material Files - Android file manager with the Material You design, can also be used as an FTP server or an SMB/SFTP client.

RCX - Android port of Rclone that can connect to supported cloud providers.

SD Maid SE - Powerful file management tool for Android that specialises in freeing up space and removing unwanted data.
Note: SD Maid SE isn't feature complete as it is an in-progress reimagination of SD Maid to work better with newer phones.

Video/Music

xManager - Manage and install all versions of the Spotify app, unlocking features such as Ad-block and skipping tracks.

Dentex YouTube Downloader - Allows users to download YouTube videos and extract/convert audio.

Seeker - Unofficial Soulseek client for Android.

Seal - Audio/Video Downloader GUI for use with yt-dlp's supported platforms.

MPV-Android - A free media player that supports a wide variety of media file formats, audio and video codecs, and subtitle types.

Torrenting

LibreTorrent - Open Source Torrent client for Android.
LunaSea - Usenet/Torrent manager. Useful for those who utilize seedboxes.

Mods

ReVanced - Android App patcher that provides patches to popular apps such as YouTube, Twitter and Reddit.
Note: It requires the need of a microG client for YouTube and YouTube music if your device does not have root access. At the moment, the most compatible microG manager with ReVanced according to the community is Vanced MicroG, but it may be replaced in the future.

Vendetta - Cross-platform plugin-based Discord (React Native - above version 126.21) client mod.
Aliucord - Plugin-based Discord (Kotlin - below 126.21) client mod for Android. Note that Aliucord only supports versions before the React Native update (all versions before 126.21)

Miscellaneous

Lucky Patcher - Android app modifier, can be used to cheat, hack or crack Software.
This app isn't open source so giving it root access to use some of it's features is your responsibility!