Static electricity is a buildup of electric charge on the surface of a material. The charge stays put until it can be discharged - either gradually or suddenly (like a spark).
Current electricity is the continuous flow of electrons through a conductor. It has two major subtypes:
In the late 1800s, Thomas Edison championed DC while Nikola Tesla and George Westinghouse pushed AC. AC won because transformers can easily change its voltage - allowing power to be sent at very high voltage (low current, less energy loss) over long distances, then stepped back down for safe home use. DC can't do this as easily, though modern technology (high-voltage DC, or HVDC) is now making DC transmission more practical again for certain applications.
In summary: There are two core types - static (charge at rest) and current (charge in motion). Current electricity divides into DC (one direction) and AC (alternating direction), with AC powering the world's grids and DC powering most electronics. The
SparkFun AC vs DC guide and
Electricity Forum overview are good references for further reading.