A dialect is simply a regional variation of a particular language. In the United States, for example, there are differences in words, phrases, and syntax in New England states, in Appalachia, in the Midwest, and in the South. Even within each of those regions, there are still more local variations. While I am not acquainted with dialects in languages other than English, I am quite sure they exist as well. A dialect should not be confused with an accent, which is a different way of pronouncing words.
Before we talk about intelligibility, we should discuss why there are dialects to begin with. Some dialects exist because the ethnic origins of people differ. For example, many people who settled Appalachia were from particular sections of the British Isles, so while they spoke English, they spoke a particular form of English that remained part of their language culture. Some dialects exist because people who move to a different location encounter new flora, new fauna, and new situations. They develop their own words and expressions to communicate about these. Some of the most colorful expressions in English are a result of people settling in a new place. Until modern times, people could not travel as they do now, had no computers, radios, or televisions, and so were more isolated in their geographic "pockets." The more isolated people are, the more their language will vary from the "standard." But today, as we are all less isolated because of the ease of travel and the influence of the media, the more at risk our dialects are. That is a shame because dialects do enrich a language.
Now, insofar as intelligibility is concerned, there are enough common elements in the various English dialects to allow us to communicate fairly well with one another. If I am speaking to someone from the South, though, that person might use a word or expression I am not familiar with, so I need to ask what it means. Similarly, in my part of the United States, people use some words and expression someone from another section might not understand. If you watch television or listen to the radio, though, you will notice that most people speak what we call Standard English, which avoids the use of dialect completely. Since national radio and television programs are meant to be heard coast to coast, people in all regions of the United States need to understand what is being said. If local words, expression, and syntax are used, only people in a particular area will be able to understand.
I should also add that when there is sufficient time and isolation, a dialect can become another language. As mankind has spread across the planet, new situations and isolation have created the world's variety of languages.
The history of language and dialect is fascinating, and a wonderful area of study. Presently, there is a dictionary project, which is not complete, but which provides regional language information for the United States. This is called the Dictionary of American Regional English(DARE), and I believe there are volumes available up through the letter "R." I have included a link to the website, in case you are interested in learning more about this, and I would imagine most public libraries have copies of the completed volumes.