A mild history

Curiosity will land in a region where this key item on the checklist of life’s requirements has already been determined: It was wet. How could have Mars been wet? Mars axis isn’t stable like Earth, and when it tilts extremely the poles grow resulting in ice ages.
Gradual changes in Mars axial tilt results in ice ages. Image: NASA/JPL-Caltech
This ice cycles back to liquid water during periods of warming. Here is NASA’s hypothesised history of water on Mars:
Image: NASA
Today we still find evidence of ice below the surface when asteroids strike as seen by the Mars reconnaissance orbiter
Ice exposed by an impact crater. Image: NASA/JPL-Caltech
Except this ice will evaporate rapidly because water isn’t stable on surface due to the low atmospheric pressure. Below is some recent evidence of current outflows. It is believed that these streaks are formed by short term discharge of salty waters when Mars heats up briefly in the summer.
Short term discharge captured by MRO. Image: NASA/JPL/Univ. of Arizona
If we look at ancient Mars using this Topographic map, we see enormous outflow channels and valleys into blue depression.
Topographic map of Mars. Image: NASA/JPL-Caltech
So water wasn’t “short term” in the past as it is today. There was definitely a watery past.

Where are we looking (spatially)?

Running water results in sediments which are deposited in an alluvial fan. Here is the Landing site in relation to the alluvial fan fed by Peace Vallis:
Alluvial fan fed by Peace Vallis. Image NASA
Using an IR spectrometer, we can better understand the composition of the sediment carried by the water:
IR spectroscopy reveals the distribution of clays (green) and salts (pink). Image: NASA
Green have been identified as clay and carbonates. This is important as clays are a result of long term interaction of water and rock. While the pink represents salt minerals (sulphate) which are deposited by water. Here is a close up picture showing the border of Columbus Crater. Sulfate salt deposits ring the crater like a bathtub ring and were deposited after the clays, as the lake dried out.
"bathtub ring" of salts around the edge of Columbus crater. Image NASA/JPL-Caltech/MSSS/JHU-AP
This means that the crater was once filled with water. Below is an artist's rendition of a hypothetical sea that may have once filled Mars’ largest crater, Hellas, located in the planet’s southern hemisphere.
Hypothetical sea on Mars in Hellas crater. Image: NASA
If we look at the history of Martian rock more broadly we notice we see 2 phases in the geologic record: “iron/magnesium clays” and then “aluminum clays”.
Example of a geologic record on Mars. Image: NASA/JPL-Caltech/Univ. Arizona
Iron/Magnesium clays form when the ratio of water interacting with rock is low. While aluminum clays are signs of a high water/rock ratio as soluble elements are carried off by water which alters the composition of the rocks.  Aluminum clays may form by near-surface leaching while iron/magnesium clays may form in the subsurface.
Impact craters are excellent locations to explore rock history, so long as they date back to the periods when Mars could have been much wetter…

Where are we looking in time?

Martian valley networks have long been viewed as some of the best evidence of prolonged surface water on Mars. The density and complexity of the networks makes it much more likely we're looking at valleys that formed by runoff of some kind of precipitation. Below is a global map of valley networks on Mars, based on a global mosaic of Mars Odyssey THEMIS images. You can see how the valleys follow local topography
Global map of valley networks on Mars. Image: NASA/JPL-Caltech/ASU/Brian Hynec
These new results imply that Mars had a long-lived period or periods of mild conditions toward the end of the Noachian epoch that supported a hydrologic cycle and potentially a biosphere. Most of these systems seem to have formed around the Noachian-Hesperian boundary (3.8–3.6 billion years ago). That's a very narrow span, at a very ancient time.
Mars geologic history. Image: NASA
So we hope to examine the history of rock at key divisions in the geological record. right around the “clay era”, which is wet, then dry which results in sulphate deposits.

Next we choose our destination….