Open source

What does open source mean and why should you care about it?

Free as in free beer
Free as in freedom of speech
Vendor lock-in
Exchanging your data

Office-suite

LibreOffice is a free and open-source office suite, comprising of programs for word processing (Writer = [equivalent to] Word [in Microsoft Office]), the creation and editing of spreadsheets (Calc = Excel), slideshows (Impress = PowerPoint), diagrams and drawings (Draw), working with databases (Base = Access), and composing mathematical formulae (Math = Formaula Editor). It, overall, has similar functionality to programs included in Microsoft Office and (from version 6.2) a similar user interface, aiming to make the transition from Microsoft Office easy. LibreOffice is available for all major platforms, including Microsoft Windows, macOS, and Linux. In addition, there is a LibreOffice Viewer for Android, as well as an online office suite LibreOffice Online.
LibreOffice uses OpenDocument file format (ODF; an international ISO/IEC standard) as its native format to save documents for all of its applications. It supports the file formats of most other major office suites, including Microsoft Office. However, the import from and the export to Microsoft office doesn't always work flawlessly. This is - in my opinion - more a problem of the Office file format because it can't be either taken for granted that, e.g., a Word document created on a Mac opens identically on a Windows PC when using this format.

Software to create and manipulate graphics

Pixel-based

GIMP

Vector-bases

Inkscape

Statistical analyses

PSPP is intended as open source equivalent and replacement for the proprietary program SPSS (you can download them free-of-charge). However, even though basic functionality is implemented, quite some more advanced methods (notably ANOVA for repeated measurements) are currently missing.

R is a programming language and free software environment for statistical computing and graphics supported by the R Foundation for Statistical Computing. It is the most comprehensive and most advanced software package when it comes to statistical analyses. The disadvantage is that - even though there are graphical user interfaces - it is mainly command-line-based and therefore subject to a certain learning curve. However, you are rewarded for this effort by a wealth of included (and additional) libraries that provide a wide variety of statistical and graphical techniques, including linear and non-linear modelling, classical statistical tests, time-series analysis, classification, clustering, and others. The R community is very active in contributing or extending packages. These contributors are often experienced statisticians and therefore R and its libraries usually represent state-of-the-art-techniques for data analysis, visualization, etc. R and its libraries are provided with open source licenses (meaning that you can download them free-of-charge, can check the code and can adjust it to your needs).

Jamovi and JASP are graphical user interfaces and aim to provide a similar look-and-feel as the SPSS user interface (to ease transition). The development team for Jamovi was part of the JASP team before they split. As a consequence, both software packages are very like each other and provide generally very similar functionality. Advantages of JASP over Jamovi are (a) Bayes statistic is a central part of the JASP (however, it is possible to run Bayes analyses in Jamovi as well using an additional module); (b) APA-style formatting of output tables; (c) the option to export output tables in LaTeX format and (d) context-help (brief introduction what the statistical procedure does). Jamovi has the following advantages over JASP: (a) syntax can be used (i.e., you can take the analyses that you clicked together in the GUI, copy and adjust them so that you can directly use them in R); (b) in-program-data-editing (JASP opens data in an external spreadsheet-program like Excel or LibreOffice Calc); (c) JASP is more recent, therefore based on programming technology that wasn't available when JASP was written and therefore a bit snappier.
The package can be downloaded free-of-charge: Jamovi and JASP.

Danielle J. Navarro has written an excellent textbook covering a wide range of statistical methods that can also be downloaded for free: the original book was for R; before it was adapted for Jamovi (by David R. Foxcroft) and for JASP (by David R. Foxcroft and Thomas J. Faulkenberry). Danielle also started working on a more comprehensive version using R and tidy.

All software packages mentioned above are available for the three major operating systems (Windows, Mac OS X, Linux).