We'll keep updating this list of free and useful astronomy software available from some outstanding people and companies. These are favourite ones that we use or have used in the past.
DeepSkyStacker - Excellent free stacking software. Also offered is DeepSkyStacker Live, which builds the final image frame by frame as they are being downloaded
EQASCOM - A telescope driver that provides the astronomical 'brains' of the mount control system. This can replace your hand controller and take telescope alignment and control to the next level, including using wireless game controllers for mount movement and focusing to building complex alignment lists. Requirements: a PC running Windows, the ASCOM Platform installed, EQASCOM installed, a compatible mount controller, a mount interface, cabling. A very popular choice for Skywatcher EQ type mounts and others. Generally you will need something like a ShoeString Astronomy EQDIR Interface Module (DB-9 connector) and a serial to USB cable to connect to your PC.
GIMP - A free alternative to Photoshop
PHD Guiding 2.0 - Simply the best free autoguiding software, and possibly the best in any category - Thanks Craig!
Stellarium Astronomy Software - Stellarium is a free open source planetarium for your computer. It shows a realistic sky in 3D, just like what you see with the naked eye, binoculars or a telescope