The first fonts used in computers were designed to be installed and viewed on a desktop or laptop computer to may be get printed later. Those are called system fonts or desktop fonts. Today, system fonts are stored in a TrueType format.

Several desktop font formats have been used for print publishing and general personal computer use for the last 20 years. The major formats of this kind are:
Bitmap fonts. suitcase
First technology to produce highly legible fonts on screen. Those fonts were named after cities (New York, Chicago, Geneva). Where stored when not used in a "suitcase" inside the system. There were different files for each font-size. They worked through a technology called QuickDraw.

Adobe Type 1 (Mac postscript version )     True Type (mac)		True Type (PC)     Postscript (PC)
	] Open type [

PostScript
The first high quality fonts to be used in personal computers. Created by Adobe but adopted by Apple computers, made it possible to scale type to any size while preserving smooth curves and sharp edges. MORE

TrueType (name.ttf)
The most common format in use today. Created by Apple (to solve the problems of early bitmap fonts by outlining the shapes in vector graphics what allows perfect/easy scaling) licensed later by Microsoft. Both companies made this font format the standard for their operative System fonts.
They use simpler internal mathematics to describe their letter forms, relying on straight lines and less-advanced bezier curves. MORE
Uses one file -a suitcase- to contain all its output instructions for print and screen.
Fonts were still stored in the System file but could now be installed using drag-and-drop.

Current look of a TrueType suitcase (without an extension)
Naming of a TrueType file (name.ttf) Different members of the typeface family in different files. .ttf extension

OpenType (name.otf)
Open standard, created by Adobe and Microsoft to include more sophisticated features.
Any OpenType font uses a single font file for all of its outline, metric, and bitmap data, making file management simpler.
It makes it possible to pack thousands of character variants or glyphs into a single typeface file. You can also use the same font file on Windows and MacOS X and it includes typographic capabilities like true small caps, better support for non English glyphs or better support for ligatures. MORE, and MORE

Web open font Format (WOFF)
WOFF is essentially a wrapper that contains sfnt-based fonts (TrueType, OpenType, or Open Font Format) that have been compressed using a WOFF encoding tool to enable them to be embedded in a web page.
The format uses zlib compression, typically resulting in a file size reduction from TTF of over 40%.

True Type and Open type fonts
Linking to industry-standard TrueType (TTF) and OpenType (TTF/OTF) fonts are supported by Mozilla Firefox 3.5+, Opera 10+,Safari 3.1+, Google Chrome 4.0+.
Internet Explorer 9+ supports only those fonts with embedding permissions set to installable.

Adobe Typekit
Font hosting. It requires creating an account and paying a monthly fee. More HERE

Google Fonts API
Free service of web font hosting. Thousands of fonts to choose from. SEE
Link to 25 outstanding Google web fonts HERE

Web fonts or downloadable fonts
Font downloading for displaying web text is included in CSS3 "fonts module".
B y using a specific CSS @font-face embedding technique it is possible to embed fonts such that they work with IE4+, Firefox 3.5+, Safari 3.1+, Opera 10+ and Chrome 4.0+.
Web fonts allow Web designers to use fonts that are not installed on the viewer's computer.
Some commercial foundries object to the redistribution of their fonts.

Font hosting services allow users to pay a subscription to host non-Web-safe fonts online. Most services host the font for the user and provide the necessary @font-face CSS declaration.