In more detail
Density is ρ = m / V, in kg/m³ (SI) or g/cm³; 1 g/cm³ equals 1,000 kg/m³ or about 62.43 lb/ft³. Water is about 1.000 g/cm³ (densest near 4 °C), ice 0.917, aluminium 2.70, iron 7.87 and gold 19.32. Most materials expand when heated, so density falls as temperature rises, and the density of a gas also depends strongly on pressure. Relative density (specific gravity) divides a density by that of water and has no unit.
Why it matters
Density decides whether something floats: objects less dense than water float, which is why ice sits on top of a lake. It also converts between volume and mass, for example when ordering concrete by the cubic metre or turning a recipe's cups into grams, and it helps identify materials.
Example
Archimedes' test on a ring: it has a mass of 10.0 g and displaces 0.52 cm³ of water.
ρ = 10.0 g / 0.52 cm³ = 19.2 g/cm³
That matches gold (19.32) and rules out silver (about 10.5). Likewise 54 g of aluminium filling 20 cm³ gives 2.7 g/cm³.
Common confusion
Density is not weight: a kilogram of feathers and a kilogram of lead have the same mass but very different volumes. See Mass and weight.