camera calibration: why chessboards?
This is mostly a matter of history. 

Checkerboard patterns have proven in the past to:


yield calibration accurate results
be easy and robust to implement.


This was confirmed (for example) by a benchmark performed by NASA long time ago, where they were assessing visual solutions to control robotic arms.

Why are checkerboards robust and accurate? Because they provide features that can ve computed as intersection of lines.
Linear features were easy to detect accurately (using Canny-Deriche, SUSAN, Hough transform...), while most point detectors were computed with pixel accuracy. 
Furthermore, checkerboards do provide line intersections, which are:


robust: if a part of the calibration pattern is occluded, you can still get the coordinates of the intersection points thanks to the line equation
accurate: line equations can be interpolated from sample points, and this is the case for their intersection too. Thus, you naturally get subpixel precision. This holds also for the old-fashioned pipelines where the corners were actually clicked by somebody


By robust to occlusion, I mean that you just have to continue the parts of the lines that are visible to get the intersection coordinates.
With dots this is more hazardous, because you have less clues to infer the overall line direction. Also, when some dots are occluded, it's very difficult to know which ones are indeed occluded. Instead, using checkerboards of known size (e.g. $8 \times 8$ squares), then you can infer which tile exactly is hidden as long as you see the borders of the checkerboard.

A possible advantage of checkerboard patterns is that since you have line featurs, then it may be easier to infer distortion coefficient values, which is harder to do if you need to "connect the dots" of aligned points.

Modern point detectors (starting from Harris corners) perform interpolation to reach subpixel accuracy in point detection, and since checkerboards provide a lot of corners they continued to be used, just switching from line primitives to corner primitives.