The Reynolds number (NRe) can show how likely it is that the stream goes turbulent. Laminar flow is layered flow where parallel components of the stream do not mix: if you inject ink into the stream, like Reynolds did, you see it follows a smooth streamline. The viscous forces of the fluid layers on another are strong enough to keep the layers together despite the chaotic irregularities in the flow channel. You can see this effect, if you try to remove dust on the car with a garden hose and it doesn't work. The viscous force of the water on the surface is strong enough to form a layer where the water is still. The water coming from the garden hose doesn't have enough speed to break the smooth sticky layer on the surface. It can be derived that, in a pipe, the velocity distribution is nicely parabolic when the flow is laminar.
When the chaotic disturbances win, turbulent flow results. Ink injected to the stream disperses laterally in a smoke-like pattern. The kinetic energy of the flow is so large that the "bouncing-off" effects of inertia become more important than the "sticky" effect of viscosity. A useful analogy is car suspension: with enough speed, the suspension can't keep the car from vibrating. The velocity distribution is more even: the velocity at the surface is larger than in laminar flow, because the fast-flowing fluid away from the surface mix with the slow-flowing fluid near the surface.
To design a number to measure this effect, we need to take a look at a system where a viscous fluid flows. First, there is a term to define: the characteristic length. For a flat plate, it is its width (orthogonal to the flow). For a pipe, it is the diameter. For a conduit of some other shape, like a trough, it is the hydraulic diameter: four times cross-sectional area by wetted diameter. (It's no accident that the hydraulic diameter of a pipe is its diameter.)
The inertial forces depend directly on the kinetic energy of the flow, which is a function of the density (ρ) and the square of velocity (v2). The forces are measured per characteristic length (L). The term becomes:
ρ·v2
Inertial forces ~ ----
L
The viscous forces depend directly on the viscosity (η) and the velocity (v). The fluid layer sticks to - exerts a viscous force upon - an area of the pipe, which is a linear function of the characteristic length L. (For example, the area of a cylinder is π × diameter × length.) Viscous forces are also measured per characteristic length, so the term for L is squared - one for the dependency on area, and one per the definition "per characteristic length".
η·v
Viscous forces ~ ---
L2
When the effects of inertial forces are dominant, turbulence is more probable; and when the viscous forces dominate, turbulence is weak. The Reynolds number, which measures the turbulent effects, is their ratio.
Inertial forces ρv2/L ρvL
NRe = --------------- = ----- = ----
Viscous forces ηv/L2 η
Because this number is a ratio of forces to forces, it is dimensionless and no unit conversions get in the way. Flow is usually always laminar below a NRe of 2100. There is a transition region between 2100-4000, where the flow may or may not be turbulent, depending on unpredictable apparatus details. NRe greater than 4000 is a sure sign of turbulent flow, except in very special cases.
The Reynolds number has more uses than just finding out how turbulent the flow is. For example, calculating the flow friction factor ξ - which is needed to calculate the flow resistance loss in net positive suction head - requires NRe.
For the reason why you would want a dimensionless number, see Buckingham pi theorem.
Sources:
Cooling Zone: Dimensionless Numbers in Heat Transfer: http://www.coolingzone.com/Content/Library/Tutorials/Tutorial%201/DNHT.html
Geankoplis, C.J.: Transport Processes and Separation Process Principles. Prentice Hall.
Keskinen, K.I.: Kemian laitetekniikan taulukoita ja piirroksia. Otatieto.