Across mathematics we find various and sundry flavors of algebra. Many of these algebraic objects, like monoids and groups and rings, relate to each other very precisely, which motivates mathematicians to study these algebraic theories as mathematical objects. Operads are one of the simplest mathematical models of algebraic theories. They only describe structures made up of operations with n inputs and 1 output, along with equations between them with all distinct variables in the same order like (x + y) + z = x + (y + z), with an emphasis on how operations compose with one another. In this talk I'll define operads, show through examples how they express familiar algebraic concepts like associativity and units, describe variations on operads that can handle commutativity and inverses, and give an overview of my research on using operads to model algebraic theories for cell diagrams.