System Specs
General
The current specific requirements to run the game are not well known, and subject to change. As of 09/18/2012, the paid version (0.16) is somewhat higher spec than the demo (0.13.3). Performance tuning can be expected as time passes.
Better hardware helps, but sometimes in non-obvious ways: -A fast (per core) CPU. Dual-core may help, but given the limitations of the Unity engine, more will do very little. -A dedicated graphics card (desktops & larger laptops)
The same is true of spacecraft design: -Fewer parts, especially struts. -Avoid highly detailed 3rd party parts. -Keep installed part counts low. -Use map view at low altitudes to avoid looking at terrain.
Demo (0.13.3)
Fast Pentium 4 (>3 GHz) Fast shader model 2.0 card (3.0 or later recommended) that does not steal from system memory. Roughly an ATI Radeon 9xxx or nVidia FX/Gefore 5 series. 768 MiB of RAM
This is the absolute minimum to get a playable (if slow) game with smallish rockets, however. A faster system will greatly enhance one's experience.
Paid (0.16)
Core2Duo 1 GiB of RAM (2 recommended. 4+ if you want to use many mods.) Dedicated video card. (Though some people have reported adequate performance with onboard video)