Intuition behind the ricci flow
See
J. Hyam Rubinstein and Robert Sinclair. "Visualizing Ricci Flow of Manifolds of Revolution", Experimental Mathematics v. 14 n. 3, pp. 257–384. (Journal link)
(Image from that paper, via Wikipedia)
I've combined two answers into one (out of order).
Second answer: some historical intuition. This is only a partial answer. Assume that you have stumbled upon the equation $\frac{\partial}{\partial t}g=-2\operatorname{Ric}$ and that you are interested in whether you can use it to deform metrics to better metrics on closed manifolds.
PDE intuition. The first question is that of short time existence given a $C^{\infty}$ initial metric $g_{0}$. So one linearizes the operator $g\mapsto-2\operatorname{Ric}_{g}$ and computes its symbol and finds that it is weakly elliptic. In fact $\operatorname{Ric}_{\varphi^{\ast}g} =\varphi^{\ast}\operatorname{Ric}_{g}$, accounts for the kernel of the symbol. By breaking the diffeomorphism invariance of the Ricci flow in the right way, DeTurck simplified Hamilton's proof of short time existence by obtaining an equivalent equation for the metric which linearizes to a heat-type equation.
To see if the metric gets better, one computes the evolutions of geometric quantities associated to $g(t)$. If $Q=Q[g]$ is such a quantity, assuming the variation $\frac{\partial}{\partial t}g=-2\operatorname{Ric}$ of the metric, one computes the corresponding variation $\frac{\partial Q}{\partial t}$. One immediately sees heat-type equations everywhere. For example, the scalar curvature evolves by $\frac{\partial R}{\partial t}=\Delta R+2|\operatorname{Ric}|^{2}$. Since the global method of the maximum principle relies on local calculations, it applies to closed manifolds. So $R_{\min }(t)=\min_{x\in M}R(x,t)$ is nondecreasing. One finds other examples of Ricci flow preferring positive curvature over negative curvature. Basically, any polynomial of the curvature and its covariant derivatives, whether it be a function or more generally a tensor, satisfies a heat-type equation. E.g., derivative of curvature estimates follow from the maximum principle.
Having obtained some control of the metric as it evolves, one then aims to prove convergence. In dimension two, this is always possible after rescaling to normalize the volume to be constant. Generally, an Einstein metric shrinks, is stationary, or expands according to whether $R$ is positive, zero, or negative, respectively.
Quantities satisfying heat-type equations. The full curvature tensor $\operatorname{Rm}$ satisfies an equation of the form $\frac{\partial }{\partial t}\operatorname{Rm}=\Delta\operatorname{Rm}+q(\operatorname{Rm})$, where $q$ is a quadratic polynomial. Since $\operatorname{Rm}$ is a symmetric bilinear form on the vector space $\wedge^{2}T_{x}^{\ast}M$ at each point $x$, we have the notion of nonnegativity of $\operatorname{Rm}$. Since $q(\operatorname{Rm})$ satisfies a property sufficient for the maximum principle for systems to be applied, $\operatorname{Rm}\geq0$ is preserved under the Ricci flow. Generally, we can analyze the behavior of $\operatorname{Rm}$ by the maximum principle under various hypotheses.
Geometric application. In particular, when $n=3$ and $\operatorname{Ric} _{g_{0}}>0$, we have $\pi_{1}(M)=0$ and hence the universal cover $\tilde{M}$ is a homotopy $3$-sphere. Encouraged by this, Hamilton proved that the solution to the normalized Ricci flow exists for all time and converges to a constant positive sectional curvature metric; thus $M$ is diffeomorphic to a spherical space form. The main gonzo estimate is $\frac{|\operatorname{Ric}% -\frac{R}{3}g|^{2}}{R^{2}}\leq CR^{-\delta}$ for some $C$ and $\delta>0$. Intuitively, we expect $R\rightarrow\infty$ and hence $\operatorname{Ric} -\frac{R}{3}g\rightarrow0$.
Singularities. Schoen and Yau proved that if an orientable $M^{3}$ admits a metric with $R>0$, then it is a connected sum of quotients of homotopy $3$-spheres and $S^{2}\times S^{1}$'s.\ Yau proposed to Hamilton that in this case Ricci flow should be able to produce surgeries to obtain a connected sum of spherical space forms and $S^{2}\times S^{1}$'s. One first sees, that by the strong maximum principle, the universal cover of singularities of the Ricci flow often split as products of $\mathbb{R}$ with a solution on a surface. This is a motivation to study the Ricci flow on surfaces, to rule out the formation of the cigar soliton.
Inspired by his corresponding results for the curve shortening flow and the Ricci flow on surfaces, Hamilton proved that the Li-Yau differential Harnack method extends to the Ricci flow assuming $\operatorname{Rm}\geq0$. Since $3$-dimensional singularity models have $\operatorname{Rm}\geq0$, Hamilton was able to classify certain singularities as steady gradient Ricci solitons and cylinders. Provided the Little Loop Lemma is true, or more aptly, no local collapsing is true, for finite time singular solutions one obtains round cylinder $S^{2}\times\mathbb{R}$ limits unless one is one a spherical space form. At this point, one can begin to believe that Ricci flow does indeed perform the surgeries that Yau proposed.
First answer: Ricci flow as a heat-type equation. Remark about Hamilton's statement: "The Ricci flow is the heat equation for metrics". (The following is a well known calculation.) The Ricci tensor is given in local coordinates by \begin{align*} -2R_{jk} & =-2\left( \partial_{q}\Gamma_{jk}^{q}-\partial_{j}\Gamma_{qk}% ^{q}+\Gamma_{jk}^{p}\Gamma_{qp}^{q}-\Gamma_{qk}^{p}\Gamma_{jp}^{q}\right) \\ & =-g^{qr}\partial_{q}\left( \partial_{j}g_{kr}+\partial_{k}g_{jr}% -\partial_{r}g_{jk}\right) +g^{qr}\partial_{j}\left( \partial_{q}% g_{kr}+\partial_{k}g_{qr}-\partial_{r}g_{qk}\right) \\ & \quad\;+\left( g^{-1}\right) ^{\ast2}\ast\left( \partial g\right) ^{\ast2}\\ & =\Delta\left( g_{jk}\right) -g^{qr}\left( \partial_{q}\partial_{j}% g_{kr}+\partial_{q}\partial_{k}g_{jr}-\partial_{j}\partial_{k}g_{qr}\right) +\left( g^{-1}\right) ^{\ast2}\ast\left( \partial g\right) ^{\ast2}\\ & =\Delta\left( g_{jk}\right) -g_{k\ell}\partial_{j}\left( g^{qr}% \Gamma_{qr}^{\ell}\right) -g_{j\ell}\partial_{k}\left( g^{qr}\Gamma _{qr}^{\ell}\right) +\left( g^{-1}\right) ^{\ast2}\ast\left( \partial g\right) ^{\ast2}. \end{align*} In harmonic coordinates $\{x^{i}\},$ $0=g^{ij}\Gamma_{ij}^{k},$ so then $-2R_{jk}=\Delta\left( g_{jk}\right) +Q\left( g^{-1},\partial g\right) ,$ where $Q$ is quadratic in both arguments. (From line to line, various terms are absorbed in the lower order quadratic term.)
In normal coordinates $\{x^{i}\}$ centered at $p$ we have $g_{ij}\left( x\right) =\delta_{ij}-\frac{1}{3}R_{i\ell mj}\left( p\right) x^{\ell}% x^{m}+O\left( r^{3}\right) $, where $r=d\left( x,p\right) =(\sum_{i}% (x^{i})^{2})^{1/2}$. Then $\Delta\left( g_{ij}\right) \left( p\right) =-\frac{2}{3}R_{ij}\left( p\right) $. Note that $\partial_{i}g_{jk}\left( p\right) =0$.
Hamilton likes to joke that when he first wrote down the Ricci flow equation, he wrote: $\frac{\partial}{\partial t}g_{ij} = 2 R_{ij}$, having a preference for positivity over negativity.
December 13, 2013. Answer to Qfwfq's question. $\partial g$ denotes a nonspecific factor of the form $\partial_{i}g_{jk}$. $\ast$ denotes a product, possibly together with contractions (summing over a pair of repeated indices, one upper and one lower). For example, \begin{align*} 2\partial_{i}\Gamma_{jk}^{\ell} & =\partial_{i}(g^{\ell m}(\partial_{j} g_{km}+\partial_{k}g_{jm}-\partial_{m}g_{jk}))\\ & =g^{\ell m}(\partial_{i}\partial_{j}g_{km}+\partial_{i}\partial_{k} g_{jm}-\partial_{i}\partial_{m}g_{jk})\\ & \quad-g^{\ell p}g^{qm}\partial_{i}g_{pq}(\partial_{j}g_{km}+\partial _{k}g_{jm}-\partial_{m}g_{jk})\\ & =g^{\ell m}(\partial_{i}\partial_{j}g_{km}+\partial_{i}\partial_{k} g_{jm}-\partial_{i}\partial_{m}g_{jk})+(g^{-1})^{\ast2}\ast(\partial g)^{\ast2}. \end{align*}
The original idea behind the Ricci flow equation, namely $$ \frac{\partial g}{\partial t}=-2Ric (g)$$ was to deform 'rough' or 'uneven' metrics to try to obtain more uniform ones. Heuristically, we can see that regions where $Ric>0$ tend to contract under the flow, thus making curvature even more positive (think for example of a sphere with the usual metric; this has constant positive curvature, so the metric will contract uniformly. In this case the sphere shrinks, thus making curvature even larger, until it finally becomes a point). In contrast, regions with $Ric>0$ tend to expand, thereby reducing the curvature. As is obvious from the case of the sphere, in general the volume $V(t)=\int d\mu(g)$ changes with time. We can renormalise the flow to make volume constant in time (incidentally, this renormalized Ricci flow was the original flow that Hamilton used in his 1982 JDG paper). This yields, at least in principle, a method to try to deform a given metric into a 'more uniform one' (i.e. one with constant sectional curvature, Ricci flat, Einstein, etc.). In practice this is not always possible because sometimes certain regions of the manifold shrink to fast and the curvature goes to infinity in finite time. This is called a singularity. These singularities in the flow are important because they give us information about the underlying topology and the geometry of the initial manifold.
Elaborating a little bit more on Prof. Agol's comment, another way to get some feel for the Ricci flow is to think of it as a heat equation. In normal coordinates, the Ricci tensor can be expressed as $-2R_{ij}=\Delta (g_{ij})+2Q(g,\partial g)$, (ie. the second term only dependes on $g$ and its first derivative). On these coordinates the Ricci flow looks like $$ \frac{\partial g_{ij}}{\partial t}=\Delta (g_{ij})-2Q_{ij}, $$ which looks suspiciously similar to the usual heat equation (it is not exactly a heat equation, though!). From this we can see that regions where curvature is big and positive tend to shrink (because the $g_{ij}$ have a value smaller than average in such a region; hence $\Delta (g_{ij})<0$), whereas regions with very negative curvature tend to expand (in this case $\Delta (g_{ij})>0$).
Finally, it is worth saying something about the asympotic behaviour of the flow. In the paper cited before, Hamilton showed that a simply connected 3-manifold $(M,g(0))$ with $Ric(0)>0$ converges eventually to a manifold of positive sectional curvature (more precisely, $g(t)$ converges in some sense to a metric $g_\infty$ of constant sectional curvature; the convergence is a rather technical issue so I will not elaborate more). In general, provided that we do not encounter a singularity in the way, the Ricci flow converges to something called Ricci solitons, which are generalizations of the more well known Einstein metrics.
Details about the asymptotic behaviour of the flow are endless, but I would recommend to have a look at section 3 of Hamilton's 1995 paper, The formation of singularities in the Ricci flow, for lots of intuitive examples. Also chapter 3 in Peter Toppings excellent Lectures on the Ricci Flow provides an introduction to the maximum principle, which is a great tool to understand this evolution equation.