Note:
Typekit is now Adobe Fonts
We used to sync fonts to a computer to use them in desktop software programs; now we activate them.
The fonts are still added through your Creative Cloud desktop application–which is explained in this tutorial–and include the same font licensing for personal and commercial projects. Learn more about Adobe Fonts.
When you activate fonts from Adobe Fonts, they will appear in the font menus of all your desktop applications, such as Photoshop, Illustrator, InDesign, Microsoft Office, and iWork. Use these fonts for print design, website mockups, word processing, and more.
Fonts are activated on your computer through the Creative Cloud desktop application. If you do not have the software installed, download it from https://creative.adobe.com/products/creative-cloud.
Start by browsing the library of fonts. You can toggle between the default & Japanese font collections from the mode selector at the top of the page.

Add filters to refine the list of fonts. You can filter by classification (such as serif or sans serif), properties (x-height, width, weight), or language.
The collection of Japanese fonts has different filtering options, so you can browse for kana-only fonts or by style classifications like Mincho, Gothic, Maru Gothic, and Brush.

When you find a font you like, click the "Activate fonts" button to add the whole family to your computer. You can also click on the font family name to open the family page, and select individual font weights or styles to use.

Active fonts will be listed in the Fonts tab of the Creative Cloud desktop application on your computer.

If your fonts don't show up in the Creative Cloud desktop application within a minute or two, check the troubleshooting guide.
Once the fonts are active, they are added to the font menu in each application, alongside all of your installed fonts. They will be immediately available in most programs, but a few need to be restarted to add new fonts to the menu (for example, Adobe Acrobat and Microsoft Office).

In addition to being listed in the Creative Cloud desktop application, your active fonts are listed on the website under the Active Fonts tab on My Adobe Fonts. Read more about managing your fonts.