Graph theory is the study of graphs, which are mathematical structures used to model pairwise relationships between objects.
The core ideas of graph theory that relate to programming are:

Nodes - Objects/entities in a graph, represented as vertices
Edges - Relationships between nodes, represented as lines connecting vertices
Paths - Sequence of edges that connect two vertices
Cycles - Closed paths where the start and end vertex are the same
Connectivity - How "connected" a graph is, aka how easy it is to get from one vertex to another

Some ways this relates to programming:

Nodes could be data objects, and edges could be relationships/dependencies between them
Finding paths/cycles could be used for routing/navigation
Checking connectivity could be used to detect disconnected parts of a graph, which could indicate error conditions
Graphs are commonly used to model networks, and programming the algorithms to analyze graphs has many applications in networking, data structures, algorithms, etc.
Many real world problems can be modeled as graphs, so graph theory gives us a toolkit of algorithms and ways of thinking to solve these kinds of problems.

So in general, graph theory is a way to formally reason about and solve problems involving objects and their relationships, which comes up very commonly in programming.
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