Multicore
One processor with two or more independent cores. The cores read and execute ordinary CPU instructions (add, move, branch) at the same time, which helps programs that can run in parallel.
Examples from the original note: AMD A-Series, Athlon II, FX-Series, Phenom, and EPYC; IBM POWER4, POWER5, and POWER6; Intel Core 2 Duo, Core i3, Core i5, and Core i7.
Manycore
A specialist multicore design for a high degree of parallel work, with a large number of simpler cores (tens, hundreds, or thousands). Used in embedded systems and high-performance computing.
Examples from the original note: the Sunway TaihuLight supercomputer, GPUs, Intel Xeon Phi, Tilera, and the Teraflops Research Chip.
Archive note, October 2017. Newer writing here is on VMware Cloud Foundation, Avi, and NSX.