Tropical geometry can be described as "algebraic geometry" over the semifield $\mathbb{T}$ of tropical numbers. As a set, $\mathbb{T}=\mathbb{R}\cup \{ -\infty\}$; this is endowed with addition being given by the (usual) maximum of real numbers and multiplication by the (usual) sum of real numbers. Recently there has been a lot of research on this kind of geometry. The obvious question is now: why are people interested in this?
There are three possible motivations I am aware of:
(1) "Polynomial" equations over $\mathbb{T}$ can be interpreted as linear equalities and inequalities over the classical reals. So people working in linear optimization and similar areas can use tropical geometry as an alternative, more algebraic approach that might lead to new methods and insights.
(2) In quantization jargon, one can interpret $\mathbb{T}$ as a "classical limit" of semifields which are all isomorphic to $\mathbb{R}^{\ge 0}$ with the usual addition and multiplication. More precisely, for any $q>0$ set $\mathbb{T}_q=\mathbb{R}\cup\{ -\infty \}$ and consider the bijection $\log_q:\mathbb{R}^{\ge 0}\to\mathbb{T}_q$ (setting $\log_q(0)=-\infty$). Pushing forward the usual semifield structure on $\mathbb{R}^{\ge 0}$ along $\log_q$, we get a semifield structure on $\mathbb{T}_q$. Then it is easy to check that the semifield $\mathbb{T}$ is, in the obvious sense, the limit of $\mathbb{T}_q$ as $q\to 0$. So if one likes to think in these terms (I do not, but that shall not bother me for now), then real algebraic geometry appears (with a grain of salt) as the quantum version of tropical geometry, which in turn gives tropical geometry an important rôle.
(3) This is only a post hoc justification. Some problems of classical algebraic geometry, mainly in enumerative geometry, have been solved by methods of tropical geometry. The usual strategy is as follows: we have a question in algebraic geometry, to which the answer is supposed to be an integer (say). We then set up an analogous question in tropical geometry, prove that the answers to the two questions agree, and then work on the tropical question, which is usually much simpler to answer.
Besides these three arguments, do you have any other motivations for studying tropical geometry?
Best Answer
My personal story with this question is that, sometime in 2007, I wanted to find a project for a student I was mentoring at RSI (a program for high school students which produces real research) and thought some variant of the question "how can you visualize all the different geometric structures on a topological torus (elliptic curve/$\mathbb{C}$)?" would be motivational. (In the end, he went with a different algebraic geometry project and it was a great success. He is on this site and can probably guess who he is.) I mentioned this to a friend of mine, who does symplectic geometry (!), who said "isn't tropical geometry the canonical answer to that question?"
I found out that this was so. Through tropical geometry, you can represent nonisomorphic genus-one tropical "curves" as visually distinct planar graphs. I actually taught a topics course on the subject to half a dozen undergrads and learned other great visualizations: for example, the tropical version of Bezout's theorem is proven by actually counting triangles in the Newton polygon. The degree-genus formula too.
I even found out that it was practical (how practical? See this paper by Eric Katz): you can deduce actual algebraic geometry theorems from it. Of course the enumerative results of Mikhalkin are the most spectacular examples, but I was pleased to work out a deduction of Bezout's theorem over the field of Puiseux series from the tropical theorem (and, if you like nonconstructive isomorphisms, you can get the theorem over $\mathbb{C}$).
I don't actually do any research in the area, but it's very appealing.