Schwarzschild spacetime
A Schwarzschild black hole is the simplest type of black hole: it does not rotate and has no electric charge. It is named after Karl Schwarzschild, discovered in * and published in 1916.
One choice of coordinates, and probably the most common one, is Schwarzschild-Droste coordinates (t,r,θ,φ), under which the metric takes form
in geometric units G=c=1. (Droste is not usually credited, but deserves to be. See *)
History
This was the first non-trivial exact solution found to Einstein’s field equations.
Symmetries
Schwarzschild spacetime does not change over time, and is spherically symmetric. Mathematically, these symmetries are described by the following Killing vectors:
\partial_t \partial_\phi
Curvature
Christoffel symbols, and curvature tensors. Some sources giving curvature quantities in various coordinates are: Hartle §B for Schwarzschild coordinates, Frolov
Coordinate systems
Schwarzschild
G-P
E-F
K-S
Orbits: velocities and frames
Static observer. 
.
Geodesic motion. Worldlines parametrised [well, mostly…] by invariants e, the “energy per unit mass”, and 
, the “angular momentum per unit mass”.
Radial motion: Taylor & Wheeler term “rain”, “hail”, “drips”. I add a 4th metaphor, “snow”, for e≤0 which is only allowed inside the event horizon r=2M. These have zero angular momentum (
). 4-velocity 
.
More generally, $u^\mu=
Tetrad: Frolov §2.11.2 citing Luminet and Marck (1985) http://adsabs.harvard.edu/abs/1985MNRAS.212…57L