How do I cite a source that I read about in a different source?
You are referring to secondary citing. Meaning you are quoting a source that was mentioned in an article/book that you read. But you did not read the entire text of the work beyond what you found quoted. The article/book that you read is the secondary source and the information you are quoting is from the original source.
In MLA style, citations taken from a secondary source should generally be avoided; consult the original work whenever possible. If only an indirect source is available, put the abbreviation qtd. in (quoted in) before the indirect source in the parenthetical reference and include the indirect (secondary) source in the Works Cited. (MLA Style, sec. 6.4.7)
parenthetical reference in MLA Style
In a May 1800 letter to Watt, Creighton wrote, "The excellent Satanism reflects immortal honour on the Club" (qtd. in Hunt and Jacob 493).
works cited list in MLA Style
Hunt, Lynn, and Margaret Jacob. "The Affective Revolution in 1790s Britain." Eighteenth-Century Studies 34.4 (2001): 491-521. Print.
MLA citation examples from http://library.williams.edu/citing/styles/mla.php
In APA Style, citations taken from a secondary source should be used sparingly, for instance, when the original work is out of print, unavailable through usual sources, or not available in English. Give the secondary source in the reference list; in text, name the original work and give a citation for the secondary source. (from http://www.apastyle.org/learn/faqs/cite-another-source.aspx)
parenthetical reference in APA Style
Goldman and Goldman's study (as cited in Linebarger, 2001) found ....
reference list in APA Style
Linebarger, D. L. (2001). Learning to read from television: The effects of using captions and narration. Journal of Educational Psychology, 93(2), 288-298.
APA citation examples from http://library.williams.edu/citing/styles/apa.php