The processor was a significant evolution in a long line of processors that stretched back to the Intel 8080. The predecessor of the 80386 was the Intel 80286; to a 32 bit architecture of its precursor the 80386 added a better (so called flat) memory model that made the implementation of multiprocessing operating systems relatively easy. Before that time, Personal Computers based on Intel processors were almost exclusively driven by the DOS and CPM systems that could run only one program at a time.
Intel later introduced the 80486, but neither it nor its successors going by the Pentium name were as big a step as the 32-bit flat addressing made by the 80386. Most applications running on PCs in 2001 will run on the older 80386; there are only a few instructions that the later generations added and they can in most cases be easily replaced. Building a program for the 80286 is much harder, and usually requires fundamental changes to the application.
Because of large amount of compatibility the whole range of processors compatible with the 80386 is often collectively termed the i386 architecture; the instruction set for the architecture is known as IA32.
See Intel for a comprehensive list of all CPUs produced by that company.