Before we get started discussing some commonly used connectors, let's explore the terminology used to describe connectors.
Gender - The gender of a connector refers to whether it plugs in or is plugged into and is typically male or female, respectively (kids, ask your parents for a more thorough explanation). Unfortunately, there are cases where a connector may be referred to as "male" when it would appear to be female; in the examples section, we'll point a few of those out as we discuss individual component types and explain why that's the case.
Polarity - Most connectors can only be connected in one orientation. This trait is called polarity, and connectors which have some means to prevent them being connected wrong are said to be polarized, or sometimes keyed.
Contact - Contacts are the business portion of the connector. They are the metal parts which touch each other, forming an electrical connection. This is also where problems occur: the contacts can become soiled or oxidized, or the springiness required to hold the contacts together may fade with time.
Pitch - Many connectors consist of an array of contacts in a repeated pattern. The pitch of the connector is the distance from the center of one contact to the center of the next. This is important, because there are many families of contacts which look very similar but may differ in pitch, making it difficult to know that you are purchasing the right mating connector.
Mating cycles - Connectors have a finite life, and connecting and disconnecting them is what wears them out. Datasheets usually present that information in terms of mating cycles, and it varies widely from one technology to another. A USB connector may have a lifetime in the thousands or tens of thousands of cycles, while a board-to-board connector designed for use inside of consumer electronics may be limited to tens of cycles. It's important that you select a connector with a suitable life for the application.
Mount - This one has the potential for being confusing. The term "mount" can refer to several things: how the connector is mounted in use (panel mount, free-hanging, board mount), what the angle of the connector is relative to its attachment (straight or right-angle), or how it is mechanically attached (solder tab, surface mount, through hole). We'll discuss this more in the examples section for each individual connector.
Strain relief - When a connector mounts to a board or cable, the electrical connections tend to be somewhat fragile. It is typical to provide some kind of strain relief to transfer any forces acting on that connector to a more mechanically sound object than the fragile electrical connections. Again, there will be some good examples of this later on.