From Hangouts in Gmail - Hangouts Help

Turn on Hangouts in Gmail --- You can try the new Hangouts from the menu in Google Chat.

1. Open Hangouts in Gmail.
2. At the top of your Google Chat list, click your profile photo.
3. Click Try the new Hangouts.
4. Gmail will refresh, and you can start using Hangouts.

Short Answer

http://hangouts.google.com

Explanation

When this answer was originally posted, there was not a Google Hangouts standalone app. It was launched in August 2015. On the first days, the 'hangouts.google.com' redirected to 'talkgadget.google.com' but nowadays the redirection was removed.

Other ways to access to Hangouts

Besides the standalone web app, it's possible to access Hangouts from the desktop without going to Gmail or Google+ web sites:

• Chrome extension
• Chrome desktop app

References

• We are launching another way to use Hangouts today Google+ post from Jordanna Chord on August 17, 2015.
• Hangouts services to be served from 'hangouts.google.com' - Google Apps update alerts.