Digital magazine publishing is the act of making a magazine available electronically on the internet. This can be a digital version of a magazine that is already in print - for example, "Hello!" the celebrity gossip magazine is sold both in shops, and via a digital edition online.
There is usually little or no difference between the two editions other than the way they are delivered. Digital editions can be downloaded and viewed on a portable device like an iPad or Kindle.
Another example of digital magazine publishing is magazines that are produced only in digital form, and are available on the internet (often for free). These are not exactly a "magazine" in the traditional sense, and have more similarities to a standard website. But the idea and content would fall into this category. Because of this, they are sometimes referred to as "webzines".
Fiction Crowd is a great example of how the lines between magazines and website have been blurred in the digital age.