The NASCAR problem is a reference to the jumble of branding icons on websites, e.g. 3rd party sign-in/login options or sharing buttons. It is dubbed the NASCAR problem because of these clusters of 3rd party icons/brands on websites resembles the collages of sponsorship decals covering NASCAR racing cars. It's a problem because such clusters of icons/brands cause both visual noise and people to be confused (see Paradox of choice), overwhelmed or unlikely to remember individual icons, especially as the clusters seem to grow with the introduction of new 3rd party identity/profile/social sites and services.
The term was originally coined by Daniel Burka of Digg to describe the user interface for signing in to websites that support third-party authentication protocols like OpenID.(date/time/source citation needed, from Chris Messina who made this assertion). Example of an OpenID sign-in UI with the NASCAR problem of a screen overwhelmed by brands:
Jeff Atwood appears to be first to have made the comparison between brand/icon links to 3rd party social sites and NASCAR in his 2006-05-11 blog post Excess Blog Flair.