The 4-pin motherboard connection is probably one of the most confusing power connections around. The area it connects to on a motherboard is not intuitive to those who have not constructed many computers. However, the shape of the 4-Pin motherboard connector is unique, both because of its 2x2 pin design and the shape of its pins.
The purpose of this connection is to provide power to the CPU. In some rare instances, mostly involving ancient motherboards and/or processors, this may not need to be plugged in. However, most modern machines will require it. This connection can be difficult to plug in, but it is designed to only fit one way, because aligning the prongs wrong could cause damage to your computer. A visual inspection of the connector should allow you to line up the connection properly. Most 4-Pin motherboard connectors keep themselves secure with a latch, and are easy to attach and detach because of this.
In some instances, when using motherboards that support multiple CPUs, you may run across an 8-pin connection. Not all motherboards which have the potential to accept an 8-pin connections need an 8-pin connection. Some will accept the 4-pin connection. There may be drawbacks to using only a the 4-pin connection, however, so consult your motherboard manual.