Classical relativity, also known as Galilean relativity, is what you learn in high school; it has simple transformations between different reference frames, where every reference frame is on the same "time" but have different positions and speeds for different objects. And if you want to transform from one reference frame to another, you basically just add all the speeds and displacements, linearly. So if you're changing from frame A to frame B, then the speeds are all V_B = V_A + V_difference, and the positions are all x_B = x_A + x_displacement. It's nice and simple, mathematically.

Galilean relativity is really accurate in most circumstances, but not in all — specifically it becomes less and less accurate the faster you are moving. The experimental discovery of this led to the development of special relativity. Basically, physicists (including Einstein, Lorentz, and Poincaré, among others) realized that the transformation leaving time unchanged was wrong, and that the transformation laws weren't that simple. Lengths had to shorten (length contraction) and durations had to lengthen (time dilation) in order to preserve the speed of light across all reference frames. This makes special relativity a bit more complicated than Galilean relativity (it's no longer just adding speeds and displacements, you have to do multiplication and division of various extra factors now), but fortunately it's not too complicated — you can understand most everything in special relativity with just high school algebra, it's just not neat and linear anymore like Galilean relativity is. This new flavor of relativity was just called "relativity" by Einstein, and today is known as "special relativity" because it is a special case of general relativity.

Here's a really good short video illustrating the difference in reference-frame transformations between special relativity and Galilean relativity.

Not long after special relativity was developed, Einstein was able to further generalize special relativity, which lead to the development of "general relativity." General relativity is basically special relativity, but in curved spacetime (where special relativity only allows for flat spacetime). Because it can handle curved spacetime (and because it expresses a specific relationship between the curvature of spacetime and matter/mass), it can also model gravitational dynamics. Gravity then is not some extra force that needs to act on flat spacetime, it emerges naturally as a fictitious force (or "inertial force") due to the curvature of spacetime. General relativity is substantially more complicated, mathematically, than special relativity is — it involves non-linear partial differential equations, a special kind of abstract geometry called pseudo-Riemannian geometry, etc. But for modelling the most complicated gravitational situations (like black holes, neutron stars, and the cosmos as a whole) it is necessary. Special relativity and Newtonian gravity just don't give the correct answers together. General relativity became famous when it first predicted (or actually, retrodicted) the correct value for the precession of Mercury's orbit and for the deflection of light around the Sun from sources behind it.

Hope that helps!