Tropical geometry is often thought of as the algebraic geometry over the tropical semiring. A good part of it is combinatorial in nature, with relations to the (geometry and combinatorics of) polyhedra and toric geometry. Recently it found applications in explaining mirror symmetry at a more fundamental level.
Tropical algebraic geometry establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones.
Example In algebraic geometry one often work with polynomials. In tropical geometry, these polynomials are “tropicalized” and this turns them into piecewise linear functions.
For instance: $f (x, y) = x^2 + y^2-1$.
This tropicalizes to $trop(f ) = x^2 \oplus y^2 \oplus 0 = min(2x, 2y, 0)$, and this is a piecewise linear curve.
(To see this remember that in the tropical semiring, the sum of two numbers is their minimum, and their product is their sum. $x^2 = x\oplus x$, $y^2 = y\oplus y$ and so $x^2+y^2$ converts to $min(2x,2y)$. The 0 will remain mysterious for the moment. (If you cannot wait look at the AARMS notes listed below.))
Textbook accounts/lecture notes include
Diane Maclagan, Bernd Sturmfels Introduction to tropical geometry, draft book
Mark Gross, Tropical geometry and mirror symmetry, CBMS regional conf. ser. 114 (2011), based on the CBMS course in Kansas, AMS book page, pdf
G. Mikhalkin, Tropical geometry, book, early draft, scribd; Tropical geometry and its applications, Proc. Intern. Congr. Math., V. 2 (Madrid, 2006), Eur. Math. Soc., Zürich, 2006, 827–852 djvu pdf
Diane Maclagan, AARMS Tropical Geometry:lecture notes from a four week graduate summer school on Tropical Geometry held at the University of New Brunswick in July/August 2008 under the auspices of the Atlantic Association for Research in the Mathematical Sciences (AARMS).
MSRI introductory workshop on tropical geometry page, Aug 24-28, 2009 (with videos of the lectures)
Erwan Brugallé, Kristin Shaw, A bit of tropical geometry, arxiv/1311.2360
Dan Abramovich, Moduli of algebraic and tropical curves, arxiv/1301.0474
Dan Abramovich, Lucia Caporaso, Sam Payne, The tropicalization of the moduli space of curves, arxiv/1212.0373
I. Itenberg, G. Mikhalkin, Geometry in the tropical limit, arXiv: 1108.3111
M. Einsiedler, M. Kapranov, D. Lind, Non-archimedean amoebas and tropical varieties, math.AG/0408311
Mikhail Kapranov, Thermodynamics and the moment map, arxiv/1108.3472
Walter Gubler, A guide to tropicalizations, arxiv/1108.6126
E. Katz, A tropical toolkit, math.AG/0610878 (Expo. Math. 27 (2009), No. 1, 1-36)
Oleg Viro, Hyperfields for tropical geometry I. hyperfields and dequantization, arxiv/1006.3034; Tropical geometry and hyperfields, talk at Mathematics - XXI century. PDMI 70th anniversary, video; On basic concepts of tropical geometry, Trudy Mat. Inst. Steklova 273 (2011), 271–303
Patrick Popescu-Pampu, Dmitry Stepanov, Local tropicalization, arxiv/1204.6154
W. Gubler, Tropical varieties for non-Archimedean analytic spaces, Invent. Math. 169 (2007), 321–376.
David Speyer, Bernd Sturmfels, Tropical mathematics, math.CO/0408099
Andreas Gathmann, Tropical algebraic geometry, math.AG/0601322
Diane Maclagan, Polyhedral structures on tropical varieties, arXiv:1302.5372
Paul Johnson, Hurwitz numbers, ribbon graphs, and tropicalization, arxiv/1303.1543 (pages 55-72 in CM580)
Brugalle Erwan, Markwig Hannah, Deformation of tropical Hirzebruch surfaces and enumerative geometry, arxiv/1303.1340
Qingchun Ren, Steven V Sam, Bernd Sturmfels, Tropicalization of classical moduli spaces, arxiv/1303.1132
Martin Ulirsch, Functorial tropicalization of logarithmic schemes: The case of constant coefficients, arxiv/1310.6269
Luis Felipe Tabera, On real tropical bases and real tropical discriminants, arxiv/1311.2211
Simon Hampe, Tropical linear spaces and tropical convexity, arxiv/1505.02045
Tyler Foster, Introduction to adic tropicalization, arxiv/1506.00726
G. Mikhalkin, Quantum indices of real plane curves and refined enumerative geometry, arxiv/1505.04338
> We associate a half-integer number, called the quantum index, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $\pi^2$ times the quantum index of the curve. We use the quantum index to refine real enumerative geometry in a way consistent with the Block-G"ottsche invariants from tropical enumerative geometry.
Diane Maclagan, Felipe Rincón, Tropical ideals, arxiv/1609.03838
Kiumars Kaveh, Christopher Manon, Khovanskii bases, Newton-Okounkov polytopes and tropical geometry of projective varieties, arxiv/1610.00298
Alain Connes, Caterina Consani, Geometry of the scaling site, arxiv/1603.03191
Keyvan Yaghmayi, Geometry over the tropical dual numbers, arxiv/1611.05508
An alternative algebraic framework for tropical mathematics (not based on semirings), “more compatible with valuation theory” has been proposed in
Connections to diophantine integration (involving p-adic integration):
MathOverflow : Mikhalkin’s tropical schemes versus Durov’s tropical schemes, learning-tropical-geometry
$n$Cafe: Tight spans, Isbell completions and semi-tropical modules