The instrument (therefore 'i') variation of VST Plug-ins.
VSTi (aka VST Instruments) is a soft-synth plug-in interface that was developed by the German company Steinberg, originally in order to provide the ability for third-party companies to develop plug-ins for it's popular Cubase VST audio sequencing package.
The protocol enbales MIDI data flow from the sequencer to the synthesizer engine, and audio data flow in the opposite direction, as well as full automation of all playing parameters.
Platforms that support VSTi include: Logic Audio, Cubase VST, Cakewalk, Fruityloops, Orion, and much more.
Some popular VSTi plugins include: Pro 52, Battery, Reaktor, Virtual Sampler, Halion, Waldorf Attack. This is of course a very partial list.
The VSTi technology (along with similar technologies such as DXi), is one of the main reasons that many musicians now days prefer to use software only studio soloutions as oppsed to a mixture of hardware synths and software sequencing.