While this is primarily a list of sites, know that browsing a website is not the only way to look for typefaces. Some type foundries still publish specimen catalogs, and some now have mobile apps and Adobe plugins. Many will also have e-mail newsletters to update on new things.
FontShop - A great collection of fonts including all of the FontFont collection. They have a plugin to help you browse while you work and their now-digital FontBook is a really extensive thing.
Emigre - One of the pioneers of digital typography, but they've continued to make quality type. You might recognize Mrs Eaves from the WordPress logo. You can also get their catalogs for just a couple of bucks.
House Industries - Prolific, cool, unique. They have some great fonts. They distribute their well-designed catalogs for free. They also run a website called Photo-Lettering where you can buy typeset headlines at a low flat price if you don't want to pay the full price for a font you'll only use once.
Sudtipos - Their stuff is available on MyFonts, but sometimes it's nice to browse straight from the foundry. They tend to make really high-quality, script-y, hand-lettered-looking stuff.
Hoefler & Frere Jones
Chank - Mostly quirky display faces.
Font Bros - Specializing in Retro faces
Lost Type Co-Op - a foundry releasing on a 'pay-what-you-want' pricing model. One of the few places to get a really nice set of curated faces for very little money (free, if you want). The drawback, however, is their stuff tends to be over-used.
Adobe - one of the grandfathers of the type foundries.
Okay Type Foundry - Great fonts including Harriet Series and Alright Sans.
Webtype - I've found this foundry to be top quality for webfonts.
Fontsquirrel - great free webfont packages with the actual different vector versions. Also has a tool to generate webfonts from your own ttf fonts.
Google Webfonts - massive selection of CDN hosted fonts that you can also download for desktop use
Typekit - commercial webfont provider
I encourage others to edit this post and add additional links if something has been missed.
I don't think there's any typeface "commonly used by professionals." Every designer has their own preferred typefaces. And many designers simply use what's on their system and rarely, if ever, purchase additional fonts unless they are forced to.