Summary
The relational model has rigorously defined query languages that are simple and powerful.
Relational algebra is more operational; useful as internal representation for query evaluation plans.
Relational calculus is non-operational, and users define queries in terms of what they want, not in terms of how to compute it. (Declarativeness.)
Several ways of expressing a given query; a query optimizer should choose the most efficient version.
Algebra and safe calculus have same expressive power, leading to the notion of relational completeness.