So you might think: yeah but what can free software really do? Well, here are a few Free and Open Source Software (FOSS) solutions. For more information, please go to the FOSS section.
Here are a few use cases. For each case, there might be more than one FOSS solution. I list my preference first, then a few alternatives in no particular order and not exhaustively. I also list their commercial/proprietary competitors.
My recommendations are based on my personal opinion and I have no affiliation with any listed software projects. Please note that I am not distributing the below listed software, but simply linking to products' respective pages from the makers.
Office Applications and Document Editors
Word Processors
Goal: I want to write up a nice letter. I need to update my résumé. I'm writing my PhD thesis.
Solution: You need an
Word Processor (or an
Office Suite). Use
LibreOffice Writer. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Word Processors

LibreOffice Writer

OpenOffice Writer

Calligra Words

Abiword
Non-FOSS competitors: Microsoft Word, Corel WordPerfect, Lotus Word Pro, etc...
Spreadsheet Programs
Goal: I want to keep track of my expenses in a table. I need to make charts with this scientific data. I want to compare different products for which I have information.
Solution: You need an
Spreadsheet Program (or an
Office Suite). Use
LibreOffice Calc. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Spreadsheet Programs

LibreOffice Calc

OpenOffice Calc

Calligra Sheets

Gnumeric
Non-FOSS competitors: Microsoft Excel, Corel Quattro Pro, IBM Lotus 1-2-3, etc...
Presentation Programs
Goal: I need to make a presentation to our clients about our new product. I want create a slide show of my vacation pictures.
Solution: You need an
Presentation Program (or an
Office Suite). Use
LibreOffice Impress. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Presentation Programs

LibreOffice Impress

OpenOffice Impress

Calligra Stage
Non-FOSS competitors: Microsoft Powerpoint, Corel Presentations, IBM Lotus Freelance Graphics, etc...
Database Programs
Goal: I need a simple database to manage our employee information. I want a database to keep track of my stamp collection.
Solution: You need an
Database Program (or an
Office Suite). Use
LibreOffice Base. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Database Programs

LibreOffice Base

OpenOffice Base

Calligra Kexi
Non-FOSS competitors: Microsoft Access, Corel Paradox, IBM Lotus Approach, etc...
Office Suites
Goal: I need an integrated suite of office related applications.
Solution: You need an
Office Suite, which is a collection of office applications made by the same creator. Use
LibreOffice. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Office Suites

LibreOffice

OpenOffice

Calligra
Non-FOSS competitors: Microsoft Office, Corel WordPerfect Office, IBM Lotus Symphony, etc...
Text Editors
Goal: I need to edit some text files. I want to write a program. I want to modify source code.
Solution: You need an
Text Editor. On Linux, use
gedit. More information on
wikipedia. On Windows, use
Notepad++. More information on
wikipedia. Advanced users can use
Vim or
Emacs. More information on wikipedia about
vim and
Emacs. Click on the logos below for the respective official sites.
FOSS Text Editors

gedit

Notepad++

Vim

Emacs
Networking and Communications
Web Browsers
Goal: I want to surf the web, use social networking sites, view videos online, etc... Related protocols: HTTP and HTTPS.
Solution: You need a
Web Browser. Use
Firefox. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Web Browsers

Firefox

Epiphany

Konqueror

Chromium

Lynx
Non-FOSS competitors: Opera, Internet Explorer, Safari, etc...
Email Clients
Goal: I have several email accounts, I want to check/create email. I also want to manage mail from all of my accounts in one program. I want to keep track of my contacts, appointments and tasks. Related protocols: IMAP, POP3, SMTP
Solution: You need an
Email Client. Use
Thunderbird with the
Lightning add-on (also FOSS). It runs on Windows, Linux, MacOS and more. Made by the same group as Firefox. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Email Clients

Thunderbird

Evolution

Kmail / Kontact

Sylpheed

Mutt
Non-FOSS competitors: Microsoft Outlook, Apple Mail, IBM Lotus Notes, Novell Groupwise, etc...
File Transfer Clients
Goal: I want to upload files to my website. Related protocols: FTP, SFTP, FTPS, etc...
Solution: You need a
File Transfer Client. Use
FileZilla Client. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS File Transfer Clients

FileZilla

Secure FTP

LFTP

cURL

WinSCP
Non-FOSS competitors: WS_FTP, Transmit
Media Playing and Viewing
Media Players and Music Collection Managers
Goal: I want to listen to music and play video files.
Solution: You need a
Media Player. Use
VLC. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Media Players and Music Collection Managers

VLC

Mplayer

Rhythmbox

Amarok

Totem

Audacious
Non-FOSS competitors: Windows Media Player, Apple iTunes
Note: Distribution or even usage of certain FOSS software in this category are encumbered by patent issues, licensing issues, Digital Rights Management (DRM) control, encryption (CSS/AACS) and restrictive laws (DMCA). This is another facet of the commercial and proprietary world that imposes limits and restrictions the users in their enjoyment of their media. Specifically, affected products, among others, are: DVDs and Blue Ray discs (movies), MP3 files (music), any MPEG based formats including MPEG-2 (DVDs) and MPEG-4 (H264/AVC) (Blue Ray). These restrictions vary by country and by products. The almost all the technical aspects of the restrictions are no longer a factor as they have been circumvented. These restrictions only remain legal, political and economical in nature. More information:
Photo Collection Managers
Goal: I want to view my picture files. I want to transfer pictures from my digital camera to my computer.
Solution: You need a
Photo Collection Manager. On Linux use
Shotwell. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Photo Collection Managers

Shotwell

digiKam

gThumb
Non-FOSS competitors: Adobe Photoshop Album, Google Picasa, Apple iPhoto
Photography Workflow Editor
Goal: I'm a photographer and I want to edit my pictures.
Solution: You need a
Photography Workflow Editor. Use
Darktable. It runs on Windows, Linux, MacOS and more. More information on
wikipedia. Click on the logos below for the respective official sites.
FOSS Photo Collection Managers

Darktable
Non-FOSS competitors: Adobe Photoshop Lightroom
Operating Systems and Hypervisors
Free and Open Source Operating Systems
Goal: I just built myself a new computer, I need the software system to run all my applications.
Solution: You need a
Operating System. Use a
Linux distribution with
GNU software (GNU/Linux). It runs on PCs, MacOS, Embedded platforms and much more. More information on wikipedia on
Linux and
GNU. Click on the logos below for the respective official sites.
Since operating systems are more complex than applications, I have created a separate section for that.
Free and Open Source Operating Systems

BSD

FreeDOS
Non-FOSS competitors: Microsoft Windows, Apple OS X
Links