Calculus is the study of the properties of functions on the real number line. It is "the calculus of functions".

Calculus can be learned at two levels. What is often called "Calculus for Engineers" presents the ideas and results of calculus, with an emphasis on the applications. There are many new ideas in calculus, so the emphasis on applications often makes the subject more concrete, and the examples motivate the subject.

Analysis is the study of calculus using a rigorous approach. While calculus has many results which can be understood intuitively, analysis "analyzes" the subject at a foundational level. So, sometimes theorems become obscured by the extra rigour. But, there are many aspects of the subject that cannot be exlained or understood without the additional rigour.

It is like the difference between "common wisdom" and law. The first is easier to grasp, but "fuzzy around the edges". The latter tries to make everything precise.

Archimedes "invented" the concept of limits to find areas. Newton invented the Calculus around 1666, but did not publish for many years. Leibniz invented the calculus simultaneously, and also gave us the notation that we use today. Cauchy is considered to be the founder of "Analysis" and introduced the use of "epsilon-delta" proofs to begin making the subject rigorous.

The Math Archives has an excellent History of Calculus which discusses these people and more,

Calculus, 3rd edition, by Michael Spivak. published by Publish or Perish, Inc. Press

A classic text that develops the theorems of calculus from a mathematical foundations approach.

Calculus, 3rd edition by Hughes-Hallett, Gleason, et al.

The text, which was developed by the Calculus Consortium based at Harvard, will introduce you to calculus through the study of problems and examples, discussion of theoretical ideas, the use of the calculator, and through applications of the calculus.