Sky cartography software
The software listed in this category present a sky map, some in a realistic way, some in a more symbolic way. They can be used to display the sky that you will currently see above you, navigate in some areas of the sky in a quest for a good observation target, search for particular objects and even control telescopes.
The telescope control part, with the exception of Stellarium for which it's described in its own subsection below, uses the generic telescope link known as INDI. It works for absolute- and relative-pointing telescopes. Clicking on the map and asking the software for telescope slewing is quite simple and useful. Also, the telescope will report its position to the software and thus be displayed on the map, which is very useful in case of manual slewing.
Stellarium is a free planetarium for your computer. It shows a realistic sky in 3D, just like what you see with the naked eye, binoculars or a telescope. It is the perfect sky map for beginners and for realistic rendering.
Stellarium is written in C++ and Qt and is working on most operating systems.
Stellarium has the ability to control telescopes for GoTo operations. It does not check for sun-pointing or mount collision. Two modes of interaction are possible, as explained on their wiki page for telescope control. The first mode is using a telescope server, a stand-alone application interacting with the telescope device. It requires command-line interaction and manual configuration. The second mode is using the control plug-in, with its graphical user interface. It connects to the telescope devices either directly, with the serial port, or using the telescope servers from the first method.
The only advantage I see about using a telescope server instead of the direct link is that it can be distributed on the network. The telescope server communicates with Stellarium using a stellarium-specific protocol described here (source).
Here is the list of compatible telescopes with either ways.
Cartes du Ciel (CdC) / SkyChart
Probably the most used and reliable free sky cartography and telescope pointing software. Works on many platforms.
XEphem also features planetary cartography, a quite fun solar system ephemeris with top, side or canted views to see planet alignments and much more.
The XmTel package provides the INDI drivers (and an INDI server) for telescopes. It is compatible with XEphem.
Planetary cartography software
All sky cartography software above are able to depict the phase or appearance of planets of the solar system, or information about their visibility, their satellites, and so on. XEphem has a solar system view that enable visualizing the position of planets in it, easily understanding what happens at what date, or finding conjunctions.
Virtual Moon Atlas
Virtual Moon Atlas is a 13-year running opensource project featuring the visualization of the Moon aspect for every date and hour. Many tools are available, such as measuring, crater identification and so on, and on Windows with ASCOM, it can also pilot computerized telescopes on the Moon surface. See also the Sourceforge project.
XPlanet renders images of planets, possibly with their satellites, as they would appear at the given time. It can be used to create an image or to directly redraw the X root window, which contains the background image. It's not really a tool whose purpose is to find specific events like eclipses, but it can render them when you already know when they happen.