The Catalan numbers are one of the most important sequences in combinatorics, which studies the mathematics of counting and arranging finite discrete structures. Combinatorics is an essential branch of mathematics as it provides efficient techniques for enumerating quantities that would otherwise be difficult to quantify by conventional methods. The Catalan numbers have several equivalent definitions, including the recursive and closed forms. While the closed form is more straightforward, the recursive definition is more helpful in solving problems in combinatorics. Through the recurrence and bijective proofs, the Catalan numbers allow deriving various combinatorial objects. We will explore properties and patterns exhibited by Catalan objects like lattice paths, Dyck paths, balanced parentheses, rooted trees, full binary trees, multiplication schemes, and polygon triangulations. The recursive formula reveals the relationships between these structures, while bijective mappings prove their cardinalities are the same.