In this thesis, a basic introduction to Leavitt path algebras will be given. Originally introduced in 2005, Leavitt path algebras are a special version of algebras, mathematical structures where addition, subtraction and multiplication are defined, but not division. Any Leavitt path algebra arises from a directed graph, a mathematical structure which abstracts the real-life concept of networks. We will prove some fundamental properties of Leavitt path algebras and see that a directed graph and the Leavitt path algebra it generates are connected in very meaningful ways. We do this by looking at specific groups of elements of the Leavitt path algebra, known as ideals. The main result of this thesis is a theorem which states that specific ideals of a Leavitt path algebra correspond directly to specific groups of elements of the directed graph it is built from.