How do you all communicate? Are there ever times when there isn't one language that is common to everybody?

Excellent question! In my experience, English is generally used as a common language. In the past, German and Latin have served this purpose. However, this system is not perfect--think of all the ways grammar can be abused with humorous consequences. 

There are definitely times when there is not one language that is common to everybody, although those times are relatively rare. More commonly, speakers can imperfectly understand each other. For example, if you said, "What's up?", a non-native English speaker might look at you a bit funny. A non-native speaker may believe that you are literally asking what the words of that question indicate. A good friend has a funny story about the word "vowels" being mixed up with the word "bowels" to humorous effect.

Research is "written up" in journals. There is often a real problem with good papers in journals written in languages other than English not getting the attention that they deserve because of the language barrier. While most research groups are fairly international (and therefore have some understanding of several languages), translations are often needed. Many journals are translated into other languages. At universities, it is fairly easy to get a librarian or someone in the language department to translate a paper (or you can use Google Translate). Still, it is good to have an understanding of multiple languages to be able to communicate with the authors of important papers in your field who may speak limited English (although they may speak another language like Russian, Japanese, or Chinese almost natively).

Math is a language that is common to every scientist, contrary to the picture above (It looks and works the same in every country and the symbols mean the same things, too--it's really the universal language and good for making all sorts of friends from all over (for example, your computer is doing binary arithmetic)). I once had a professor who tended to hand out notes in a variety of languages, telling us to learn from the math (I think he was also trying to encourage us to learn German and French). When all else fails, many ideas can be communicated via a combination of creative miming and and math.