When applying for a tourist visa to visit Russia, what kind of
information can the Russian Embassy access about the applicant with
their US passport?

This is a full list of all documents you will need to supply:

http://www.russianembassy.org/CONSULAT/TOUR-VIS.HTM

you can also look around 

http://www.russianembassy.org

Hi.  I am more interested in what the Russian authorities, or any
foreign country for that matter, can access on their own with the
information on a US citizen's passport.  Such as background checks,
credit history or other personal information, or do foreign entities
not have legal access to such materials (other than public
information).  Thanks!

Here is some information, more on the ACLU site:
http://travel.state.gov/passport/eppt/eppt_2502.html

http://www.aclu.org/Privacy/Privacy.cfm?ID=17078&c=130

http://www.schneier.com/blog/archives/2005/03/us_electronic_p.html

http://www.privacyinternational.org/article.shtml?cmd%5B347%5D=x-347-180779

http://cryptome.org/dos021805.txt

There is no background check for tourist visas. Russian Embassy checks
your information against internal database only. Grounds for denial:

1. Violation of visa regulations in Russia
2. Deportation from Russia
3. Previous refusal of Russian visa (for any reason)
4. So-called "black list" (spies and wannabe's)

Also, you might be denied if the purpose of your trip clearly other
than tourist - Religious, Government, Business, Home-stay, etc. The
Consular officer will analyze your previous trips to Russia, their
frequency and the nature of your occupation.

Oleg
http://www.Russia-visa.com

Good point:  occupation.  It is probably not advisable to give your
occupation as "reporter, journalist, tv- or radio- anything, govt
employee, ...".  You get the idea?