The Intel Management Engine (ME) is a subsystem composed of a special 32-bit ARC microprocessor that's physically located inside the chipset. It is an extra general purpose computer running a firmware blob that is sold as a management system for big enterprise deployments.
When you purchase your system with a mainboard and Intel x86 CPU, you are also buying this hardware add-on: an extra computer that controls the main CPU. This extra computer runs completely out-of-band with the main x86 CPU meaning that it can function totally independently even when your main CPU is in a low power state like S3 (suspend).
On some chipsets, the firmware running on the ME implements a system called Intel's Active Management Technology (AMT). This is entirely transparent to the operating system, which means that this extra computer can do its job regardless of which operating system is installed and running on the main CPU.
The purpose of AMT is to provide a way to manage computers remotely (this is similar to an older system called "Intelligent Platform Management Interface" or IPMI, but more powerful). To achieve this task, the ME is capable of accessing any memory region without the main x86 CPU knowing about the existence of these accesses. It also runs a TCP/IP server on your network interface and packets entering and leaving your machine on certain ports bypass any firewall running on your system.
While AMT can be a great value-add, it has several troubling disadvantages. ME is classified by security researchers as "Ring -3". Rings of security can be defined as layers of security that affect particular parts of a system, with a smaller ring number corresponding to an area closer to the hardware. For example, Ring 3 threats are defined as security threats that manifest in "userspace" mode. Ring 0 threats occur in "kernel" level, Ring -1 threats occur in a "hypervisor" level, one level lower than the kernel, while Ring -2 threats occur in a special CPU mode called "SMM" mode. SMM stands for System-Management-Mode, a special mode that Intel CPUs can be put into that runs a separately defined chunk of code. If attackers can modify the SMM code and trigger the mode, they can get arbitrary execution of code on a CPU.
Although the ME firmware is cryptographically protected with RSA 2048, researchers have been able to exploit weaknesses in the ME firmware and take partial control of the ME on early models. This makes ME a huge security loophole, and it has been called a very powerful rootkit mechanism. Once a system is compromised by a rootkit, attackers can gain administration access and undetectably attack the computer.