What is a hydrometer? How is it used? What is “specific gravity”?

A hydrometer measures the weight of a liquid relative to the same
volume of water (i.e., relative densities). In brewing, much of
this excess weight is expected to be from fermentable and unfermentable
malt sugars. Most hydrometers measure Specific Gravity (SG), which
tells how many times heavier than water the liquid of interest is;
for example, a 1.050 SG wort is 1.05 times heavier than an equal volume
of water at 60 F. SG measurements are temperature dependent, and SG
should be measured at 60 F., as water is SG 1.0 at 60 F.

Hydrometers often come with a temperature conversion chart, but
hydrometers often are not accurately calibrated, so that water at
60F will not read 1.0. An easy way to take SG readings with a
hydrometer is to measure at room temperature, and then measure water
at room temperature and take the difference.

Some abbreviations commonly used in homebrewing relating to specific
gravity: OG, Original (wort specific) Gravity; FG or TG, Final or
Terminal Gravity (when the beer is finished fermenting).

Source: http://stason.org/TULARC/indulgence/beer-brewing/16-What-is-a-hydrometer-How-is-it-used-What-is-specific.html