"Font" used to refer to a complete set of characters of one typeface (set of characters that share a common design structure) in a specific size and style. So for a typesetter from 40 years ago, a typeface would be Times New Roman, a font family within it would be Times New Roman demi-bold, and a font within that would be 12-point Times New Roman demi-bold. Since the advent of digital media, the two terms are considered synonymous.
There is also Bitmap, TrueType, and PostScript fonts. Bitmap fonts have glyphs as, well, bitmaps, at different point sizes. Most of these have only one bit to tell the system if a specified pixel is used by the character or not. Most systems will allow you to use bitmap fonts at point sizes other than the what the font has, but results aren't pretty. PostScript fonts are the first vector-based font, used by printers that supported the PostScript language. The first iterations of Windows and Mac OS couldn't actually show them on screen, so a bitmap version of the font was still needed, Adobe Type Manager, available for Mac OS and Windows, would let you see the fonts (even anti-alias them), but current versions of Windows and Mac OS do this by themselves. TrueType was made to compete with PostScript by Microsoft and Apple, and a bitmap font is not needed. It also uses a different way of doing curves than PostScript.
In order to correct visually uneven spacing between two particular characters in a font, there is a process called kerning. It adds or subtracts space between characters.
It can be confused with letter-spacing or tracking, which refers to the amount of space between letters in a piece of text. Tight spacing usually benefits large types, but it has a subjective feeling ("fast talking like in advertising") whereas wider spacing increases legibility of small fonts, and creates an association of a more “objective voice”. In excess, the text can look affected.
Serif: Serifs are embellishments details added at the extremes of the strokes of some letters.
Times New Roman. A traditionally-styled font originally commissioned for newspaper columns, and thus designed to fit the maximum amount of text into a narrow space without sacrificing either readability or aesthetics. Was for many years the default font in most word processors, mainly due to being one the only fonts available by default on Windows, but is still a major standard.note To be specific, a 12 point font size is a standard in general writing. You would have to be crazy to write an entire book in 10-point font. Ubiquitous as a result, especially in non-professional work. Accused of being So Okay, It's Average, but it does its job with a good balance of readability and economy of space. Used in several books and newspapers. Before the advent of digital typesetting almost all British paperbacks were set either in Times New Roman or the more old-fashioned looking Plantin. Using it on a website, however, tends to brand the creator as a noob.
Georgia. Has larger characters than Times New Roman, with old-style (or "lowercase") figures. One of the first fonts explicitly designed for computers, emphasizing on-screen legibility.
Garamond and Baskerville (an updated Caslon, a British font used in the American colonies and in historic documents, like the U.S. Declaration of Independence. Still used and highly considered today). They offer formality and elegance.
Traktir is another slab serif typeface evoking the look and feel of XIX century papers and posters. It's bilingual, has both Latin and Cyrillic letters (the latter was often used in Tsarist Russia).
Computer Modern, the LaTeX font. Is a didone type (horizontal lines are thinner than the vertical parts).
Bodoni. Also a didone, this historic font is suitable for posters, headlines or logos.
Cambria Math is the default, and only shipped, font for the current Equation Editor in Word. Other fonts, such as the Times-like XITS Math, are available for download and sort of compatible with Equation Editor.
Sans Serif: While serif fonts are more usual in print, sans serifs are widely used online. There is no agreement in which of the two has better readability (facility to read text) and legibility (to recognize characters).
Calibri. In 2010 became the default typeface in, among others, Microsoft Office, substituting Times New Roman and Arial. Not recommended for use on Web sites, however, because at the same point size it is noticeably smaller than all of the sans-serif fonts that can be used as a fallback.
Eurostile. Known for its squarish round letters, most iconic in its extended (widened) form. Common on electronic equipment in the '70s and '80s. Still manages to retain a futuristic feel, though some would argue that it's slipping into Zeerust territory.
Frutiger. Originally designed for use on airport signs, it is notable for being easy to read from a wide range of angles and distances. Quite popular, it has a clean modern look.
Futura. Geometric type (therefore a very modern look) used extensively as a general purpose font.
Its Bauhaus style is good for a movie set in The Fifties and you want to show signage at a research laboratory or tables in a science textbook (even though it is considered more a font from The Thirties, where it may also be found, but more in the context of something high end, like a fine arts publication).
Gotham. Commissioned by GQ magazine to be geometrical and look "masculine, new, and fresh", though it was purportedly modeled after machine-cut signage from the mid twentieth century. Now associated with Barack Obama, whose campaigns made extensive use of it. Its use has explodedsince then; most amusingly, it is used in the new logo for DC Comics.
Here is a page about some of the differences between Arial and Helvetica. This is a 20-question quiz for telling apart the two fonts, using well-known logos designed in Helvetica and converted to Arial.
Univers. The Rival: Both were created the same year (1957) and are extremely legible. Univers has wider letter-spacing.
Lithos gives a primitive-ethnic feel. Good if you have a movie set in The Nineties and want to show a restaurant menu.
Roboto. The "Android font" as of Android 4.0 (Ice Cream Sandwich). Released for desktop systems by Google in 2012.
Segoe. As "Segoe UI", the Microsoft user interface font for Windows Vista and 7, along with Microsoft Office 2007 and 2010. A slightly-different version of Segoe is the font for Windows 8 and Windows Phone. Considered a Suspiciously Similar Substitute for Frutiger and Myriad.
Verdana. Sans serif counterpart to Georgia. Both were created by Microsoft.
Replacement Scrappy: Some felt that Ikea lost part of his identity when Verdana, in 2009, replaced Futura as the default font for their catalogs.
Script: Meant to imitate handwriting or calligraphy.
It's not even a very good font for comic lettering, for a number of reasons. One of the major ones being that the letters do not "flow" together well; they look almost as if they had been cut and pasted randomly from a comic book page like a ransom note. Another is that the capital I is only available with bars on the top and bottom, which is considered bad form outside of writing the word "I". There are a number of better comic fonts — Digital Strip and Anime Ace among them — but they lack lower-case forms, which is a big part of Comic Sans' appeal in non-comic use.
Kaufmann, Mistral, Dom Casual (which has a freehand effect similar to Comic Sans) and Brush Script. Casual script typefaces (they emulate informal handwriting). Very popular in advertising and entertaining magazines.
Dom Casual peaked in popularity in the mid-'50s to early '70s, when it was (over)used much as Comic Sans is now. Today it's most often used when a "retro" look is desired.
Segoe Print. Another font introduced in Windows Vista and Office 2007, presumably meant to displace Comic Sans as the informal/faux-handwriting font of choice.
Monospaced: Each character occupies the same amount of horizontal space.
Anonymous TrueType and Anonymous Pro. Initially designed for Macs, these got exported to other operating systems. Popular with coders and some developers, they have a very clean look. Often overlooked because of compatibility issues.
Courier New. Looks like typewriting. The font of choice for screenwriters; nearly every guide to writing a screenplay says to use this font. Also great for plain text e-mails, code, and rageface comics.
OCR-A. Bar code or credit card font. Was created in 1968 to be easily recognized by computers. It has a retro-futuristic look, so it's also used in advertising and display graphics.
OCR-B: Also for optical character recognition, but has a less technical appearance.
Lucida Console is the typeface used in the blue screen of death in Windows XP and Windows CE, as well as the default font for Notepad. It's also the only font that can replace the default one in the Command Prompt. In other platforms there is Lucida Typewriter.
Consolas, a favorite among programmers because of its clean look, especially when text anti-aliasing (ClearType in Microsoft land) is used. Strangely, it did not replace Lucida Console as the default font in Notepad.
Fixedsys is a very old console-based font. Unlike almost everything above, Fixedsys is notTrue Type; the characters are encoded as pixels rather than lines and curves. Was the default font for Notepad on older versions of Windows, and still finds use where terminal programs are involved.
Bank Gothic. A small-caps font with a simple, geometric design. Popular as a logotype for anything that needs to convey "serious business", particularly for subheadings.
Impact. Considered amateurish, is good for making lists or standing text out. White-with-black-border Impact is used in an awful lot of image memes.
AnotherLove It or Hate It font: it gets slapped on to everything despite it being specifically designed to evoke Egyptian-ness. Unlike Comic Sans, though, it at least looks decent in its own right; there just aren't many places where it's appropriate.
American Psycho: Despite what Patrick Bateman may tell you, there is not a font called "Silian Rail" — his card appears to be Garamond. Van Patten's "Romalian type" appears to be Didot. Bryce's card is Helvetica, Paul Allen's is Copperplate Gothic.
Not Invented Here demonstrates the annoying results of teaching someone to recognise poor kerning here, much like the xkcd strip above.
Paul: See this logo? There's too much space between these two letters. It's called "bad kerning". Desmond: Oh yeah, now that you mention it, I do. Desmond (looking around): Wait, it's here too... and here... and here... Desmond: It's everywhere! Paul: Now I can go.
College Humor has created two shorts based on fonts, Font Conference and Font Fight. Different fonts are personified by different actors, assuming personas suggested by the font names. Thus Comic Sans is a superhero, Wing Dings is a mental patient able to speak only using the names of symbols ("diamonds candle candle flag!"), Futura is a time traveler from the future, Century Gothic is a goth, etc.