The basic idea for Ubuntu is that the OS itself will be the same on your phone, tablet, PC, and TV; and, the apps that run will be the same as well and the design will adapt to the display being used. This makes a developer's job easier, because you write an app once and it will run on multiple devices. And, it makes the experience better for users because you don't have to set up the same app multiple times or worry about transferring app data, because it's all the same app. We are seeing similar functionality with Windows 8 right now in the app space, but the systems themselves are still separate.
Canonical's Community Manager Jono Bacon has released a new video showing off this convergence with a community-made Reddit app called Karma Machine. Bacon shows the app running on a phone, tablet, and Ubuntu PC and how the UI will adapt to each display. This functionality looks pretty compelling, especially keeping in mind the plans to allow you to dock your mobile Ubuntu device to a monitor and keyboard and have it serve as a full Ubuntu PC. In this case, when you dock your phone, any apps open would simply adapt to the new display, while underneath they are still running the same code in the same OS. Pretty impressive stuff.
Although, it should be noted that while Canonical will have the phone and tablet system ready to go with Ubuntu 14.04 in April, and it will be releasing the Ubuntu Software Centre for mobile devices when handsets start hitting the market, converged apps will not be coming at the same time. Converged apps rely on the new Mir display server and Unity 8 UI in order to work, but both of those pieces have been delayed and won't be part of Ubuntu 14.04. More likely, we'll see those updates and proper converged apps in Ubuntu 14.10 in October.
source: OMG! Ubuntu!