Graphs serve a couple of important functions in data analysis. First, while exploring data, they can help us understand our data and the relationships in it.
But data analysis isn't all about wrangling data and crunching numbers. We also must communicate our findings and convey evidence for our arguments. Graphs serve an important role in succinctly communicating complex relationships. While there's always the danger of creating graphs that aren't clear or that don't really have much informational content, well-done graphs can clarify concepts and relationships that are difficult to explain verbally.
Designing good, informative, and beautiful charts is difficult, and if you want to dive into that topic, and there's a lot of good information out there. Anything by Edward Tufte (http://www.edwardtufte.com/tufte/) is a good place to start, and his book, The Visual Display of Quantitative Information, is considered a classic in the field of data visualization design.
For creating charts...