Ferrule. A ferrule (a corruption of Latin  "small bracelet", under the influence of  "iron") is any of a number of types of objects, generally used for fastening, joining, sealing, or reinforcement. They are often narrow circular rings made from metal, or less commonly, plastic. Ferrules are also often referred to as eyelets or grommets within the manufacturing industry.

Most ferrules consist of a circular clamp used to hold together and attach fibers, wires, or posts, generally by crimping, swaging, or otherwise deforming the ferrule to permanently tighten it onto the parts that it holds.

Examples
The sleeve, usually plastic or metal, on the end of a shoelace, preventing it from unraveling (called the aglet)
The metal sleeve which is crimped to hold the eraser in place on a pencil
The metal band that binds the bristles or hair of a brush to its handle
The metal ring which holds a chisel blade's tang to its handle
In fiber optic terminations, glass or plastic fibers are bonded to precision ferrule connectors (FCs), also described as fiber channel connectors, and polished for splitting or connecting two fibers together.
The metal spike at the end of the shaft of an ice axe
The margin of a cast crown that stabilizes root-canal-treated teeth in restorative dentistry
A ferrule, in respect to dentistry, is a band that encircles the external dimension of residual tooth structure, not unlike the metal bands that exist around a barrel. This is also known as the ferrule effect. 
The bottom end of a flag stick on a golf course, which fits snugly into the hole in a cup
The plastic sleeve that adorns the bottom of a steel or graphite golf club shaft just above the club head hosel. Originally designed to protect the shaft from damaging vibrations, it is now used mainly for aesthetic purposes.
The metal band used to prevent the end of a wooden instrument or tool from splitting
The semi-circular metal band that holds in place the fibers on the frog of a bow for an instrument in the violin family
Compression fittings for attaching tubing (piping) commonly have ferrules (sometimes called olives, especially in the UK) in them to make a liquid-tight connection
A swaged termination type for wire rope
The cap at the end of a cane or umbrella as well as the ring, often crimped, sometimes pinned, that prevents an umbrella's canopy from sliding off the end when open
The portion of a cue in pool, billiards and snooker that tops the  and to which the  is bonded; historically made of ivory, now typically made of fiberglass, phenolic resin, brass or titanium.
A joint between sections of a segmented fishing rod 
A metal cap at the end of a cable housing with a hole on a bicycle or motorcycle control cable
The metal sleeve that connects a hypodermic needle to a plastic Luer taper (which then connects to a syringe or intravenous therapy tubing)
An electric wire ferrule is a metal tube crimped over stranded wire to secure it within a screw terminal, optionally with electrical insulation protecting any exposed portion of the wire not completely inside the screw terminal post.
A rubber or plastic cap at the bottom end of a walking stick.
The metal tip on a staff carrying a flag or guidon that finishes and protects the end of the wooden shaft.

Reasons for use
Some of the reasons people use ferrules include:
 To shield parts or cables from electromagnetic pulses, environmental damage, the elements, thermal factors, and more.
 To cover parts, adding wear resistance, damage protection, or packaging. 
 As a connector, to connect wires, structural devices, and systems
 To bind parts together, including bundles of wires, or cloth threads to the end of the mop, as an example.
 To act as conveyance for fluids like oil and water, or for gasses like air.

References

Hardware (mechanical)
Writing implements