Special relativity was first (1905) and deals with how motion, the perception of time and velocity are relative not absolute and dependent on the relative velocity of the observers. This includes E=mc^2, the way time is experienced differently by different observers moving at different fractions of the speed of light, the way that velocities add and thus how no particle with mass can ever go (or exceed) the speed of light, etc.
General relativity (1915) is a theory of gravity which replaces Newton's universal law of gravity (and reduces to it for large distances) and is a mathematical framework that describes how space-time is curved and bent by the presence of mass and how this structure effects the motion of particles. It is called general relativity because its solution in flat space (i.e. ones with no masses around) reduces to the equations of special relativity, thus special relativity is a "special" case of general relativity.
Thus, if people are talking about: Atomic clocks on space-ships not experiencing the same time, the twin-paradox, the inability to exceed the speed of light, the contraction of an object as it approaches the speed of light, etc. They're talking about special relativity.
If people are talking about: Space-time curvature due to a star or a planet, the bending of light around a star, planet or galaxy (gravitational lensing), the expansion of the universe, the big bang, etc. They're talking about general relativity.