<p>A quick glance at the present-day internet would seem to indicate that Adobe Flash is the obvious choice for embedding video in a web page.  Is this accurate, or are they other effective choices?  Does the choice of ASP.NET as a platform influence this decision?</p>

Flash is certainly the most ubiquitous and portable solution.
98% of browsers
have Flash installed.  Other alternatives are
Quicktime
,
Windows Media Player
, or even
Silverlight
(Microsoft's Flash competitor, which can be used to embed several video formats).
I would recommend using Flash (and it's FLV video file format) for embedding your video unless you have very specific requirements as far as video quality or DRM.