In physics, Fluids are defined as a substance that deforms (or flows) when a stress is applied on them. The discipline that studies this flow is known as Fluid Dynamics. Liquids, gases and plasmas are all different examples of fluids. 
— The first 195 words of my dissertation!
In order to simulate the flow of a fluid a mathematical representation of its current state is needed. At its most basic form a fluid can be thought of as a combination of two properties, velocity that defines how the fluid will move over time and density which defines its composition . For example, when simulating smoke in a room the density will represent the smoke particles that flow through the air and the velocities will determine how these particles will move around the room. Mathematically the velocities can be represented as a vector field that spans over the simulation area while the densities can be represented by individual particles that update their position by following the velocities. However, simulating each single particle can be an extremely expensive operation so the particles are often replaced with “a continuous function which for every point in space tells us the amount of … particles present” .