What is a boundary layer, exactly? I've been struggling with this concept for a very long time. I know about boundary layers in a very informal sense, such as in boundary layer separations at the trailing edge of airfoils and other objects immersed in flowing fluid. I also know about the boundary layer development in a pipe, as a plug flow develops into the parabolic distribution expected of a Newtonian fluid.
I've referred to Wikipedia, but I could not find a proper definition of what a boundary layer is:

In physics and fluid mechanics, a boundary layer is an important concept and refers to the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant.

I can expect the boundary layer connects points that have some common parameter, but what is it?
 A: The boundary layer is the region of a flow close to a surface, where there is a gradient of velocity between zero at the wall and the free-stream velocity ($v_\infty$), which is caused by viscosity and the no-slip condition. It is also possible to have boundary layers for other flow variables, e.g. temperature or scalar concentration.
From your question, it seems that what you are looking for is a firm definition for where the edge of the boundary layer is:

I can expect the boundary layer connects points that have some common
  parameter, but what is it?

Typically, the edge of the boundary layer is defined as some percentage of the free-stream velocity. From the same Wikipedia article:

Paul Richard Heinrich Blasius derived an exact solution to the above
  laminar boundary layer equations. The thickness of the boundary
  layer $\delta$ is a function of the Reynolds
  number for laminar flow.
$$\delta \approx 5.0 \frac{x}{\sqrt{Re}}$$
$\delta$ = the thickness of the boundary layer: the region of flow where the velocity is less than 99% of the far field velocity $v_\infty$; $x$ is position along the semi-infinite plate, and $Re$ is the Reynolds Number given by ...

