Open Source Software Solutions

When choosing which software to use, one philosophical choice you must make is if you're OK with proprietary software, or if you will prioritize open source solutions. Open source means if you're a developer, you can look at the code to make sure there are no backdoors or security issues. The most important time you want to make sure you're choosing an open source solution, is if the software is intended for online communication and/or it uses encryption. These are two times you want to be able to trust your software doesn't have a backdoor. While open source software doesn't mean a cryptographer or auditor has reviewed it, I argue it's a better base to start from, than blindly trusting a company, which often on foreign soil with a mandate to spy on foreigners.

Something else most open source fans don't mind, is that it's usually free. One advantage of choosing to philosophically support open source, is you'll likely never have to pay for software again.
I will update this post over time, but I hope to use it as a place to list open source alternatives to proprietary solutions that you're likely familiar with.

Last update: Jan 26, 2015:

Communications:

Instead of Skype or Megachat, consider Jitsi.

Browsers:

Instead of Safari or Internet Explorer, consider Firefox, Chrome, or for anonymity check out Tor Browser which is based on Firefox.

Office:

Instead of Microsoft Office or Apple iWork (Pages etc) check out LibreOffice and OpenOffice

Productivity:

Instead of Evernote, Microsoft OneNote or Google Keep, check out Etherpad, ProtectedText,  and Paperwork.

Email Programs:

Instead of Outlook, or Mail.app, consider Thunderbird

File Encryption:

Instead of Apple FileVault, PGP, or Symantic Drive Encryption, check out encfs

"Free" Email providers:

Remember, if it's "free", you're the product or service being sold. That being said, instead of using Gmail, Hotmail/Outlook.com, Yahoo! Mail, or Facebook Messenger, check out Autistici/Inventati, MyKolab, or Riseup.

File Storage:

Instead of Dropbox, Apple iCloud, Google Drive, or Microsoft One Drive, check out OwnCloud, SparkleShare, or MyKolab.

Networking:

Instead of Facebook, check out Diaspora, movim, and GNU social.

Image Sharing:

As a potential replacement for Flickr, imgur.com, or Instagram, check out MediaCru.sh

Mapping:

Instead of Google Maps, try OpenStreetMap and Marble Map

Search Engines:

Instead of Google or Bing, try DuckDuckGo and StartPage

Operating Systems:

Instead of Windows, or Mac OSX, consider: Mint Linux, Ubuntu, Debian, FreeBSD, OpenBSD, or anything listed at Distrowatch.

Router software:

While it will likely invalidate your warranty, you can replace the software on your wireless access point, consider OpenWRT or DD-WRT.