References:

A quick intro, a couple of relevant entries at Stanford Encyclopedia of Philosophy
"Modal Logic", by J. Garson. The Stanford Encyclopedia of Philosophy (Winter 2003 Edition), Edward N. Zalta (ed.).
"Temporal Logic", by A. Galton. The Stanford Encyclopedia of Philosophy (Winter 2003 Edition), Edward N. Zalta (ed.).
For more on the ‘spirit’ "Amsterdam school", visit the website of the Institute for Logic, Language and Information in Amsterdam
Modal Logic, by Patrick Blackburn, Maarten de Rijke, and Yde Venema. CUP, 2001.
Modal Logic, By A. Chagrov and M. Zakharyaschev. Clarendon Press, Oxford, Oxford Logic Guides 35, 1997. 605 pp.
Tools and Techniques in Modal Logic, by Marcus Kracht, Studies in Logic and the Foundations of Mathematics No. 142, Elsevier, Amsterdam, 1999.
History:
A nice summary of the development of Kripke semantics (and on why perhaps it shouldn’t be called Kripke semantics):
Mathematical modal logic: a view of its evolution’’, by Rob Goldblatt, Journal of Applied Logic, Volume 1 , Issue 5-6 (October 2003)
Two seminal papers on Kripke semantics by the man himself:
S. Kripke, Semantical analysis of modal logic i, Zeitschr. f. math. Logik und Grund. d. Mathematik 9 (1963), 67-96.
Semantical analysis of intuitionistic logic i,pp.92-130, In Formal Systems and Recursive Functions, Amsterdam, North-Holland, 1965.

Relational semantics has gained the place of one of the mainstay semantics for exploring mathematical properties of various formal languages. The foremost class of languages explored is the class of modal languages. These include alethic (truth, possibility, contingency, necessity, etc), epistemic (knowledge), temporal and spatial, to name just a few in the endless array of modalities studied using relational semantics. The usefulness of modal languages and closely related classes of languages lies in good part in their computational properties. It was shown that modal logics the semantics of which can be represented in the so-called guarded fragment of first-order are all decidable. Usefulness of relational structures does not end with modal operators, however. Various propositional connectives have been studied via relational structures (e.g. intuitionistic, relevant, causal, paraconsistent implication, various negations, etc.). In this tutorial we will emphasize the mathematical versatility of applications of this semantics.
The aim of this tutorial is to introduce the basics of relational semantics and give participants a taste of the plethora of its uses in modern logic. As we will see, the semantics and its related languages are used across the fields of AI, computer science, linguistic, mathematics and philosophy. We will emphasize its flexibility and capacity to model intentional phenomena, as well as the intuitiveness of this variable-free kind of quantification. The semantics will be viewed as a logical toolbox, much in the spirit of the so-called ‘Amsterdam school.
First, we will give a general intro into relational semantics and modal and other germane languages.
Then, we will explore the relational structures as a means for modelling propositional languages, and finally, we look at the place of this semantics in the general landscape of tools for logical modelling, that is, its connection to classical and other logics of interest.