Generally speaking, it means that the effect is derived from an algebraic theory (in a specific and structured way). While equations are definitely part of most theories, you can absolutely have a theory without equations, and furthermore you can define an effect with equations that isn't algebraic. The full definition of "algebraic theory" unfortunately doesn't really fit in a comment, but I did want to push back on the idea that "an effect becomes algebraic if you add equations to it".
In the effects literature, you often also see the definition that an operation (of an effect) is "algebraic" if the operation commutes with `>>=`. This definition is actually the same as the one above, just stated in a different way.
In the effects literature, you often also see the definition that an operation (of an effect) is "algebraic" if the operation commutes with `>>=`. This definition is actually the same as the one above, just stated in a different way.