What Spanish does capitalize are proper names for people, places, newspapers and magazines; abbreviations of personal titles such as Dr., the equivalent of the English "Dr.," and Sr., the equivalent of "Mr."; and the first word in the titles of books, plays, movies and similar works.
Here are the most common cases where English capitalizes that Spanish does not:
Calendar: Names of the days of the week and months of the year use lower-case letters. Hoy es martes. (Today is Tuesday.) México celebra su independencia el 16 de septiembre. (Mexico celebrates its independence on September 16.)
Composition titles: In formal written Spanish, titles of movies, books, plays and similar works capitalize only the first word and proper nouns. La guerra de las galaxias ("Star Wars"), Harry Potter y la piedra filosofal ("Harry Potter and the Sorcerer's Stone") Note: In informal written Spanish, and on book covers and movie posters, it is not unusual to see such composition titles capitalized as in English.
Personal titles: Introductory titles are not capitalized, although common abbreviations of them (such as Sr. for señor, Dr. for doctor, D. for don and Srta. for señorita) are. ¿Conoces a la señora Wilson? (Do you know Mrs. Wilson?) ¿Conoces a la Sra. Wilson? (Do you know Mrs. Wilson?) La reina Victoria fue mi abuela. (Queen Victoria was my grandmother.)
Religions: Names of religions and their adherents aren't capitalized. Mi madre es católica. (My mother is Catholic.) Estudio el cristianismo. (I'm studying Christianity.)
Ordinal numbers: When an ordinal number is used after a name, it isn't capitalized. Luis catorce (Luis the Fourteenth), Carlos octavo (Charles the Eighth)
Place names: Although the given name of rivers, lakes, mountains and other geographic features are capitalized, the place identifier is not. No vimos el río Amazonas. (We didn't see the Amazon River.) Vivimos cerca de la montaña Rainier. (We live near Mount Rainier.)
Nationality: Although names of countries and cities are capitalized, words derived from them are not. Soy inglés. (I'm English.) Prefiero los cocos puertorriqueños. (I prefer the Puerto Rican coconuts.)
Languages: Names of languages aren't capitalized. Hablo inglés. (I speak English.) Quiero estudiar alemán. (I want to study German.)