In U.S. publishing (and much English-language publishing generally) the titles of big things get underlined or, more commonly, italicized; the titles of littler things or parts of big things go in “quotation marks”. Here’s a list:
|TV series titles
|specific episodes of TV shows
Note that British English typically uses ‘single quotes’ where U.S. English typically uses “double quotes” and vice versa. For more on punctuation, see The Punctuation Guide.
Also note that some publications’ individual style guidelines violate these rules. For instance, The New Yorker uses quotation marks for all types of titles. This is true for many newspapers, as well, though less so now than in the past. I assume the reason goes back to the days of hand-set type: it was faster to grab a quotation mark and set it than to have a whole separate set of italicized type. Unless you are aware of a house style, though, these guidelines are accurate for U.S. publications.