The software displays songs your friends are currently listening to – as well as suggestions from artists.
It follows moves by other social networks such as Facebook to incorporate music recommendations into their services.
Last year, Spotify announced its own “follow” system, but the functionality is yet to be rolled out to users on mobile.
Twitter’s app – called #Music – is expected to be made available to download for Apple’s iPhone shortly.
No app has been made for users on Google’s Android or the Windows Phone platforms – but there will be a web browser-based version.
It will initially be available in the UK and Ireland, US, Canada, Australia and New Zealand, with more countries being added soon.
The app was likened to a “21st Century mixtape” unveiled on Good Morning America.
In a blog post, Twitter’s Stephen Philips explained: “It uses Twitter activity, including tweets and engagement, to detect and surface the most popular tracks and emerging artists.
“It also brings artists’ music-related Twitter activity front and centre: go to their profiles to see who they follow and listen to songs by those artists.”
He added that half of the social network’s users follow at least one musician.
“This is why artists turn to Twitter first to connect with their fans — and why we wanted to find a way to surface songs people are tweeting about.”
Ahead of the app’s release, Twitter gave several musicians early access. They included Moby, who wrote: “It’s a really interesting music resource.”
Many companies have tried to tap into the potential of social recommendation for music.
There is already one London-based Last.fm, which was bought in May 2007 by CBS for £140 million, which analyses what a user listens to and offers suggestions based on the tastes of other Last.fm members who enjoy similar artists.