Short answers, pulled from the story.
Climate proxies are preserved physical characteristics from the past that stand in for missing data when direct measurement is impossible. Scientists measure a correlated property such as stable isotopes or growth rates to infer the value of interest.
First attempts at extraction occurred in 1956 as part of the International Geophysical Year. The U.S. Army's Cold Regions Research and Engineering Laboratory used an electrodrill in 1968 at Camp Century, Greenland.
Ice cores in Antarctica can reveal climate records for the past 650,000 years. A European project drilled in central Greenland where ices were 200,000 years or more at 3029 meters bedrock.
Tree rings are wider when conditions favor growth and narrower when times are difficult due to temperature and humidity factors. Scientists use these annual rings to estimate local climates for hundreds to thousands of years previous.
Pollen density provides information on short-term climatic conditions based on weather patterns from previous months while identified plant communities offer data about relative time. Lake varves are layers of fine and coarse silt deposited annually that reflect summer temperature and winter snowfall levels.
In 2002 a report published findings by Drs. Lisa Greer and Peter Swart regarding stable oxygen isotopes in ocean coral skeletal rings. Their analysis showed peaks every twelve to fifteen years in surface water temperatures from 1935 to about 1994.