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 with 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.

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.