SpaceX uses an Actor-Judge system to provide triple redundancy to its rockets and spacecraft. The Falcon 9 has 3 dual core x86 processors running an instance of linux on each core. The flight software is written in C/C++ and runs in the x86 environment. For each calculation/decision, the "flight string" compares the results from both cores. If there is a inconsistency, the string is bad and doesn't send any commands. If both cores return the same response, the string sends the command to the various microcontrollers on the rocket that control things like the engines and grid fins.
The microcontrollers, running on PowerPC processors, received three commands from the three flight strings. They act as a judge to choose the correct course of actions. If all three strings are in agreement the microcontroller executes the command, but if 1 of the 3 is bad, it will go with the strings that have previously been correct. The Falcon 9 can successfully complete its mission with a single flight string.
The triple redundancy gives the system radiation tolerance without the need for expensive rad hardened components. SpaceX tests all flight software on what can be called a table rocket. They lay out all the computers and flight controllers on the Falcon 9 on a table and connect them like they would be on the actual rocket. They then run a complete simulated flight on the components, monitoring performance and potential failures.
SpaceX engineers perform what they call "Cutting the strings" where they randomly shut off a flight computer mid simulation, to see how it responds.
Dragon uses a similar triple redundant system for its flight computers.
Bonus: Dragon V2 control panels are modified 17" Tesla Model S touchscreens and include the same Nvidia Tegra SoCs. They are not part of the flight hardware and are only used to display the GUI.
Source: Discussion with various SpaceX engineers at GDC 2015/2016
Update: After four years, SpaceX software engineers have confirmed Chromium and JS powered Crew Dragon's displays on-orbit during a Reddit AMA.