Additive synthesis is a synthesis technique that uses multiple sine waves as its source of sound. Adding more and more waves (partials) means more harmonics and different sound timbres.
While in subtractive (which is maybe the most common type of synthesis) we have an OSC base with its harmonics and then we subtract elements mainly with the filters, in additive we start to build the harmonics from scratch. A nice thing to find out about the concept of additive and the difference with subtractive is this - at least I understood many things in my first steps as sound designer. Take your favorite subtractive synth and use only one sine wave. Then try to close the cut-off frequency of the Low Pass filter. There will be no difference in timbre unless we come to the point where the Cut-Off meets the fundamental place of the sine in the spectrum - the sine wave is very pure and includes only one harmonic. With other types of OSC like Saws or Pulses we already have harmonics that affect the timbre in higher places of the spectrum, that's why the filter's influence is more perceivable.
Adding partials from other inharmonic spots of the frequency map is leading to more complex and interesting sounds.
We can then make more things like apply modulation to the harmonics characteristics for both frequency and amplitude , modulate in time the relative position of them with many ratios, manipulate their number or phases with various modulation sources or morph the macro harmonic image of them with different shapes. Each different synth has its own tools.
Generally the additive synthesis sounds are bright and fairly dissonant due to the whole process of design. Bells, plucky keys with brilliance, hollow body synths, spacious pads and experimental tonal percussions are some of the sounds that we can achieve easily.
Some of the most notable additive software synths are Loom by AIR Music Technology, Razor device of Native Instruments Reaktor and Arturia's Synclavier. There are also some multi purposed synths that included additive synthesis within their tools - Arturia Pigments Harmonic Engine is a shiny example of this. Hammonds and other tonewheel organ keys are using additive synthesis method in their core also.
Additive Synthesis offers a huge field for sonic experimentation, sometime with unpredictable results and for me they shine in high mids and high frequencies. The latter however is my personal POV and my sound design approach.
Harmonic Keys for Arturia Pigments is a freebie soundbank by Vicious Antelope that designed with additive synthesis, You could listen to its sounds in the video below.