*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*}