There was actually a project done on the definition of what a minimalist shoe is and the result was
"Footwear providing minimal interference with the natural movement of the foot due to its high flexibility, low heel to toe drop, weight and stack height, and the absence of motion control and stability devices".
If you are looking for a simpler definition, this is what Wikipedia says,
Minimalist shoes are shoes intended to closely approximate barefoot running conditions.1 They have reduced cushioning, thin soles, and are of lighter weight than other running shoes, allowing for more sensory contact for the foot on the ground while simultaneously providing the feet with some protection from ground hazards and conditions (such as pebbles and dirt).
One example of minimalistic shoes would be the Vibram FiveFingers shoes which look like this.