Sharrows (formally, "shared-lane markings") are on-street pavement markings that are intended to remind people in cars to expect people on bikes on a street. They help to indicate bike routes, and they are useful to help people on bikes position themselves in the road and encourage them to "take the lane."
Sharrows are used where streets that aren't wide enough for dedicated bike lanes. In some configurations, there may be a bike lane on one side of the road in one direction (often up a hill as a "climbing lane") and a series of sharrows in the other direction.
If you Google this topic, you'll find lots of commentary against sharrows. Too often they are used badly: Painted in locations that put people on bikes too close to parked cars, put on streets that are easily wide enough for a dedicated bike lane, or painted on streets with speed limits over 25 MPH. European bike advocates who prefer separated bike paths tend to cringe at the use of sharrows in the US; they prefer separate bike lanes and paths to mixing bikes in with cars.
Where sharrows should be used
NACTO* guidelines indicate that sharrows should be placed in the center of the travel lane on streets with speeds at 25 MPH or lower. There's lots of detail in the update to the 2012 AASHTO** guidelines (see the video here; the sharrow discussing begins around 28:50), which indicate that sharrows are to be placed in the center of travel lanes when lane widths are less then 13 feet or where there is not enough room for side-by-side operation of a car and bicycle in the lane. The Cape Cod Commission also recommends sharrows as "a benefit on narrower streets where bikes and motor vehicles must share space.
* National Association of City Transportation Officials
** American Association of State Highway and Transportation Officials
AASHTO 2012 Bike Guide video: https://www.youtube.com/watch?v=B0LW1-yXZMc
Cape Cod Commission, Connecting Town Centers to the Regional Pedestrian & Bicycling Network on Cape Cod, July 2014: http://www.capecodcommission.org/resources/transportation/Townctrconnect3.pdf