First, we should note here the difference between mass and personal communication. An email, a letter, a business memorandum, an instant message, or any document intended for a limited readership is not an example of mass communication. The definition of "mass communication" refers to some form of communication that reaches a significant proportion of a population. The enabling technology of mass communication was print, and especially newspapers, which could reach a large portion of the literate people of a nation. Subsequently, radio and television created a genuine mass audience. In the United States, certain television programs claimed over half the potential audience at a given time slot.

The internet has contributed to fragmentation of mass media. In the past, one could argue that mass media built a sense of community as millions of people simultaneously sat down to watch the evening news, sporting events, or other popular television shows. This mass audience first began to fragment with video recording technology which allowed two major changes to viewing habits, the possibility of time shifting and the ability to skip over advertising.

The internet intensified this trend of fragmentation, with many people watching video context over streaming services at their own convenience. It has also allowed the proliferation of a far greater number of channels and of people to search independently for individual pieces of content rather than packages of paid channels, a phenomenon described as "cord cutting" which has led to rapid decline in cable television subscriptions. 

The fragmentation of mass media in one sense creates a new degree of flexibility in that people have infinitely more choices of what to watch or read. Unfortunately, it also has the negative effect of encouraging tribalism, in which people live in media bubbles that echo their own preconceptions, with, for example, people of a give political disposition only viewing "news" shows slanted towards reinforcing their own current biases. 

In addition to pohnpei397's many valid points, I would offer some more technical and psychological effects;

• Computers have allowed for a greater degree of personal "customization" in communication, while simultaneously fostering certain memes that are nearly infectious in their prevalence. For example, documents can be automatically translated, or read aloud, by software, yet almost everyone with an iPhone uses the same emojis.
• Speaking of phones, many people are unaware or uninterested in distinguishing the fact that cellphones are really just handheld computers. The fact that we continue to call them phones, implying we place primacy on that function, is a bit disingenuous and might suggest that computerization has infiltrated our methods of communication more than we want to admit.
• And speaking of emojis, computerized communication has increased the rate of information exchange around the world, which amplifies certain patterns of thought while dampening others. For example, you may notice that when a certain product gains a strong foothold, such as Google Maps, other products on other platforms will often change their format to a similar one. 
• The use of computers has revealed more about the development of the human brain and body. For example, research into internet bullying has suggested that anonymous computerized communication, where accountability is low, tends to increase selfish behavior. 

Ray Kurzweil, a science writer, has published a number of insights and theories on the subject. One prediction that seems likely to me is the inevitability of many forms of communication being "handed over" to computers, since they can do so without the delay of speech, text interpretation or other processes involved in transforming a physical stimulus into a neurological signal. 

Computers (and the Internet) have had a tremendous effect on mass communication.  It has made more detailed and accurate communication possible across longer distances.  It has also made it possible to have mass communications that are more focused on individuals.

Before computers, it was much less possible to send large amounts of information rapidly across long distances.  It was possible to send large amounts of information by mail and smaller amounts could be sent by fax, but neither of these methods of communication offered anything like the capabilities that email and other computer/Internet communications offer.  Today, it is possible to send things like blueprints or maps immediately to any part of the globe.  This has constituted a revolutionary change in mass communications and has helped to globalize the world.

It is also possible to have much more focused communications today.  People like marketers can send out mass communications, but computers allow them to be tailored to individuals.  The computers allow companies to identify different segments of a population and send them individualized communications.  This allows for a blend of mass and individualized communication.

In these ways and others, computers have changed mass communication in our modern world.