There are several reasons this message pops in your application or website. This article tries to clarify each particular situation.
64-bit Windows and Office
On 64-bit Windows and 64-bit Office (2010, 2013) environments, there are many reports on this error. The fix or workaround is a bit strange but seems to work for most people out there.
The "Microsoft Access Database Engine 2010 Redistributable" installation package seems the natural one to use but several reports says it does not work.
Instead, using the "2007 Office System Driver: Data Connectivity Components" seems to solve the above problem for most people.
Other situations when this error occur?
Please send in additional setups and solutions on this issue. I'll edit this article with additional sections.