Kerbal Space Program is a space physics game created by Felipe Falanghe, developed and published by Squad.
Developed as an early-access title, game modifications are not only allowed but encouraged. The game uses the Unity3D framework, and modifications can be created using MSVS or Monodevelop/Xamarin. You don't need to buy Unity3D (though it comes with advantages), and the free version of MSVS will work.

I have a small collection of minor mods available on Curseforge here.
Many more mods, tools and links to useful sites can be found on the KSP forums here.

VisibilityToggle is a simple mod that takes control of the numpad-5 key, and uses it to toggle the visibility of most parts. Does not currently work on engine shrouds or asteroids. Development ongoing.

CursorDeleter was made in response to a Youtube video by the user Danny2462. It removes the custom cursor that was implemented in KSP 1.0.0, and resets it to the system's default cursor. This should mean you can keep whatever cool custom cursor you already have as a system default (like for instance, DAnny's Super Hilarious, Indestructible, Near-Godlike Assassins Creed cursor) without KSP overwriting it.

Old mods
Punish The Lazy introduces a reputation penalty if you spend long periods of game-time doing nothing. Ideally you should be fulfilling contracts in the Kerbin/Mun/Minmus system while waiting for transfer windows, and time-warping all the way to Jool should incur a hefty reputation penalty.
You might find that TriggerAu's Kerbal Alarm Clock will go well with this mod.
1.2.2 update is on the to-do list

HeatWarning was created for 1.0.0, to display a simple thermometer over parts that are getting close to their maximum temperature.
Since then it's been expanded upon and improved, with audio warnings and overrides, a critical heat alarm, map view/IVA effects, config menu, and the ability to disable the 1.0.2 stock indicators when the plugin is enabled.

ThrottleMemory is a still-pretty-WIP attempt to have the throttle remember where you set it before a time warp. I'm told it doesn't play well with some other mods, so YMMV.

MusicPauser mutes the music channel when the game is paused. This doesn't yet silence mods like Chatterer (or even HeatWarning), but it's useful enough in its current state for release.

ResetSAS is entirely deprecated. Despite this, it continues to get downloads. In 1.0, KSP now replicates the behaviour of ResetSAS (and Crzyrndm's SASReset) in stock, and resets your autopilot mode to stability assist when you turn SAS off.
TL;DR: If you're running 1.0 or greater, you don't need this mod.