A graph is a fancy math word for a network. This is not to be confused with the other use of the word "graph" which can refer to a plot made by graphing a function.
A network graph is composed of "nodes", also called verticies, and "edges", also called links or connections.
The Exaptive Studio and Cognitive City makes it easier for non-programmers to build graph-based data applications. Because of the interconnectedness that can be captured by a graph, these applications can be used in a variety of use-cases where traditional applications tend to fall short.
It helps to understand a little about graphs as you use the software, and how storing your information in a graph is different than storing it in a spreadsheet or relational database of related tables. Read the blog post by founder, Dave King, about why graphs are so much better than tables for keeping up with your evolving mental model of complex phenomena.