Celestia
Celestia is a free real-time space simulation that lets you experience our universe in three dimensions. Unlike most planetarium software, Celestia doesn't confine you to the surface of the Earth. You can travel throughout the solar system, to any of over 100,000 stars, or even beyond the galaxy. All travel in Celestia is seamless; the exponential zoom feature lets you explore space across a huge range of scales, from galaxy clusters down to spacecraft only a few meters across. A 'point-and-goto' interface makes it simple to navigate through the universe to the object you want to visit.
Add-ons, Mac OS X and Linux versions are also available. Don't miss the screenshot.

Orbiter
«ORBITER is a realistic space simulation using proper physics for the modelling of planetary motion, gravitational fields, free space and atmospheric flight etc. As such, it is more challenging to master than an arcade game - you should at least have some basic idea about mechanics and planetary motion to get the most from it.
You can land your ship at a space port on a planet surface, or dock to an orbital station; High resolution planetary maps provide some nice visual effects; Configurable environment. Users can add planets, space ships and space stations to the existing universe, or design a completely new solar system from scratch.»
The two minimum installation files will include the «Base» and «Textures» packages. Additional Earth, Moon, Mars etc... texture files and patches are also available. Don't miss the screenshot.

OpenUniverse
Simulates the Solar System's bodies in 3D. It does so in real time, meaning that you can view all the planets, moons and spaceships move along their paths, trace, follow, orbit and even control them (time and spaceship parameters control). Pretty realistic, it quite hardware intensive.
If you have a good graphical accelerator you may wish to download additional/optional planet textures featuring an even higher resolution.
You will need to have a few OpenGL libraries already present in you system (you probably have them without noticing it); otherwise you will need to separately install them.
There is also a Linux/Unix version available. Must see the screenshot.

Mars Explorer, Mars3D, Venus3D & Earth 3D
«Mars Explorer» allows you to fly at thousands of km/h or drive at 50km/h over an accurately rendered Martian landscape covering an area of 1000kmx1000km. The surface is drawn to scale using altimetry data from NASA's MGS satellite. The program is available in two versions: with the Labyrinthus Noctis map (LN) & with the Melas Chasma map (MC)
«Mars3D», another program by the same author, renders the whole surface of Mars in three dimensions using data collected by NASA's Mars Global Surveyor satellite. The data gives a maximum horizontal spatial resolution of 13 Km. The vertical accuracy is to within 5 m. Real time visualization of the red planet is allowed. The user has full control of the colours and amount of detail used to draw the surface. Zoom in an out features are implemented. As of the release of version 3, Mars3D is now split in several files (program, datasets and textures). Visit the homepage for details.
«Venus3D» does the same, using data from Magellan satellite, with a detail of 20 Km, for Venus.
They will run better on a fast machine loaded with lots of memory and a graphical accelerator. Highly entertaining.
«Earth 3D» is also available. You must see the screenshot.

Sky 3D
This program aims to take you in a 3D tour of our galactic neighborhood. The user sets its own way, simultaneously being presented with information on the features of the stars he is observing. See a screenshot.

Planet's Orbits 1.5
According to the author, this «is an accurate digital orrery with a wealth of functionality. Uses time-dependent orbital elements for the calculation of planetary positions (except for Pluto). Good accuracy is thus obtained for the time range of +/- 4000 years. The most noteworthy functions are date specification (past, future), 3D orbital animation and custom zoom, orbits of 7000+ asteroids, and statistic and information windows.»
We found it the best orrey drawing program ever. A huge bunch of optional data files with orbital elements are available for download. See a screenshot.

Partivew
«Partiview (Particle View) is a program that visualizes and animates three dimensional particle data. Within Partiview, you have the ability to navigate through 3-D data, as if you're "flying." Partiview is not limited to particle data; it can display 2-D images as well as multiple 2-D polygons that can be built into 3-D surfaces. It cannot record flight paths, though it can play them. You can learn more about Partiview and how to visualize your own data in the manual. In addition, visit the Digital Universe site to download the Milky Way Atlas and other data sets as they become available.»
See a screenshot.

Other freeware programs in this category (links to the homepages)

Planet Engine

3D Starmapper

3D Planetary Simulator

Free Virtual Galaxy

Other shareware programs in this category (links to the homepages)

StarStrider