TITLE: What do Algebra and Calculus mean?
QUESTION [40 upvotes]: I sometimes see phrases like 'the relational algebra' or 'the lambda calculus'. What is the difference between an algebra and a calculus?

REPLY [2 votes]: I came here hoping for an elaboration on what I found in these publicly accessible slides, but actually I think it's stated more succinctly and satsifyingly (to my mind, and limited knowledge), so I'll restate it in case it's of use:
Algebra - Procedural method/algorithm specifying how to obtain the results.
Calculus - Description of what makes a result such, without knowing how to obtain them.
The slides are in the context of the Relational Calculus (as compared to RA), so for example, given the problem of finding those people with fathers named John:
$$\pi_{\text{name}}\ \sigma_{\text{fathers_name}=\text{'John'}}\ \text{People}$$
is a relational algebra, it tells us how to find such peoples' names.
$$\{\ t\ \vert\ t\in\text{People} \wedge t[\text{fathers_name}]=\text{'John'}\ \}$$
is a relational calculus, it tells us what makes an answer; not how to find them.