At its base level, Shiny is an R package that brings R to the web.
Shiny is based on a reactive programming model, similar to a spreadsheet. Spreadsheet cells can contain literal values, or formulas that are evaluated based on other cells. Whenever the value of the other cells change, the value of the formula is automatically updated.
Shiny apps behave the same way. However, unlike a spreadsheet which requires a spreadsheet program to use reactively, a Shiny app is simply a web application created in R. With Shiny you can make your data analyses reactive and accessible to anyone with a web browser, without having to know anything about web programming.
Learn more about how Shiny works at shiny.rstudio.com