Alternative set theories
Peter, there are a couple of good survey papers on precisely this topic. Neither is quite exhaustive, but they provide good overviews (naturally, there is some overlap), and the second in particular also has an extensive bibliography:
MR2732673 (2012c:03152). Apostoli, Peter; Hinnion, Roland; Kanda, Akira; Libert, Thierry. Alternative set theories. In Handbook of the Philosophy of Science, vol. 4., “Philosophy of mathematics”, Andrew D. Irvine, ed., 461–491, Elsevier/North-Holland, Amsterdam, 2009,
and
MR3409865. Holmes, M. Randall; Forster, Thomas; Libert, Thierry. Alternative set theories. In Handbook of the History of Logic, vol. 6, “Sets and Extensions in the Twentieth Century”, Dov Gabbay, Akihiro Kanamori, and John Woods, eds., 559–632, Elsevier/North-Holland, Amsterdam, 2012.
I am more familiar with the second one, so let me say a few words about its contents:
The paper begins by discussing (Section 2) Simple Type Theory, Mac Lane's and Zermelo's, not because they are alternative theories, but because they are needed to understand some of them (such as New Foundations and its variants), though the authors mention that "Zermelo set theory or variants of Zermelo set theory have been pressed into service themselves as alternative set theories, presumably by workers nervous about the high consistency strength of $\mathsf{ZFC}$."
Section 3 covers theories with classes: First Von Neumann-Gödel-Bernays and Kelley-Morse set theory, then Ackermann set theory (where non-set classes can belong to other classes, this theory is equiconsistent with $\mathsf{ZF}$), and then a weak system that they call "Pocket set theory", an expansion due to Holmes of a suggestion by Rudy Rucker.
Section 4 covers theories with atoms and with anti-foundation axioms. They first discuss $\mathsf{ZFA}$, then Aczel's anti-foundation axiom, and Boffa's axiom (in this system, there is a proper class of $x$ with $x=\{x\}$, while in Aczel's system there is only one).
They continue in section 5 with New Foundations and related systems (such as the much better understood $\mathsf{NFU}$, where urelements are allowed). Naturally, this section occupies the main bulk of the paper.
Section 6 discussed Positive set theory and its fragments and variants, typically denoted $\mathsf{PST}$, perhaps with sub- and superscripts (an exception to this notation is the theory $\mathsf{GPK}^+_\infty$, mutually interpretable with an extension of Kelley-Morse by large cardinals). This leads to Topological set theory.
Since the systems in section 6 allow talk of super- or hyperuniveses, section 7 discusses systems motivated by non-standard analysis, such as Nelson's Internal set theory, or Vopěnka's.
Section 8 concludes the list and covers "curiosities": The double extension set theory of Andrzej Kisielewicz (that "has the property which is usually ascribed to New Foundations (we believe not entirely fairly) of being motivated by a syntactical trick without any semantic motivation"), and Zermelo's set theory extended by an axiom asserting that there is an elementary $j:V\to V$, which turns out to be significantly high in terms of consistency strength.
Let me add that the theory $\mathsf{ZF}$ augmented by such an axiom has also been studied, even fairly recently, but I would not classify it by any means as "alternative". In general, extensions of $\mathsf{ZF}$ via large cardinals, forcing, or inner-model theoretic considerations are just part of the standard set-theoretic landscape.
Something that the paper definitely does not cover is systems motivated by algebraic geometry, topos-theoretic, or categorical considerations, such as Grothendieck universes. On the topic of categorical set theory, and Lawvere’s Elementary Theory of the Category of Sets, there is a recent paper that has gathered some attention,
MR3193723. Leinster, Tom. Rethinking set theory. Amer. Math. Monthly 121 (2014), no. 5, 403–415. Also available at ArXiv:1212.6543.
(The original version of the entry on Alternative axiomatic set theories at the Stanford encyclopedia of philosophy, by Holmes, seems to have been used as a basis for the Holmes-Forster-Libert Handbook chapter. It has since been significantly revised.)
Finally, there is also Bourbaki's set theory, about which Adrian Mathias has written a few critiques. You may enjoy the discussion at the nForum.
There is the structural set theory SEAR (and variants) by Mike Shulman.
SEAR takes as basic the notion of sets, elements and relations, and has as logical grounding dependent type theory, in that elements are typed by the set they belong to (unlike ZFC, elements are not themselves sets, and cannot belong to more than one set), and relations are typed by their domain and codomain. The axioms let you construct a category of sets with nice properties very quickly (i.e. a topos) and in the end you get something equiconsistent with ZF. There is an extension SEAR-C including the axiom of choice; this is equiconsistent with ZFC.
If anyone is counting, SEAR-C has 7 axioms, two of which are schema. If I was pressed, at the moment I'd nominate SEAR(-C) for my favourite foundational system.
(Aside: People should definitely learn the not-quite formalised contrast between material and structural set theories: the former is basically anything that treats elements as supreme, having independent existence and all there is, the latter is more isomorphism invariant and elements don't have independent existence (this is my own spin on it - there is no watertight definition). Note that 'categorical' set theory is subsumed by structural set theory, as SEAR is not given by axiomatising a category of sets, it just heads that way as it was invented and developed by category theorists)
Kripke-Platek set theory is a well-explored axiomatisation of a weakened first-order set theory that gives it pleasing correlations with the language of second-order arithmetic (itself a first-order theory).
This is probably the most useful field of non-standard set theory to know if you are interested in second-order comprehension axioms, recursive model theory, or reverse mathematics.