Martin Hyland and John Power (2007). The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads. ENTCS 172:437-458.

Both monads and Lawvere theories provide characterisations of algebraic structure, with monads providing the more general characterisation. The authors provide an introduction to Lawvere theories, discusses their relationship to sets, and why monads became the more popular treatment.

Then they tackle the application of the theory to the semantics of side effects, where they argue that the generality of monads allow them to characterise computational phenomena that are not to do with side effects such as partiality and continuations, and argue that Lawvere theories more cleanly characterise what side effects are.

This paper is a good introduction to an important line of recent research done by Hyland&Power; cf. also the LtU story Combining computational effects.