The Sport:      

Modern foil fencing is the direct descendant of dueling .
Fencing combines learned reflexive skills with considered responses to observation of an opponent, applied psychology and strategy. Because these elements are all involved, but can be used indifferent combinations, a wide range of people can succeed as a fencer over a long time by exploiting their relative advantages and compensating for their relative shortcomings by thinking about what they are doing.
It is also a sport of manners. The rules require fencers to salute each other, the President and judges at the beginning and end of a bout. Dealings with the President in a bout have to be courteous.

The Weapon:       

A foil is a light, tapered, semi-flexible, tipped weapon. It is designed for attack with the point and defense, or parry, with bottom half of the blade. The blade is steel. It is oblong in section and tapered over its length to the tip so that it bends, in one plane (vertical), towards the tipped end of the blade (the foible). The thick, handle, end (the forte) remains rigid. A cup shaped guard protects the hand and fingers. The handle is long and thin with a leather or rubber sheath and a counterweight screwed on the end (a French grip) Beginners use the French grip so as to learn to manipulate the blade tip using mainly the thumb and fingers (finger work). Blades are full sized or a shorter and lighter child's blade.
For beginners' practice and competition fencing, the blade is tipped with a rubber button that can be removed and replaced as needed. This is known as a "steam" blade (i.e. not an electric blade).
The Clothing:            
Fencing is very safe. Properly supervised and conducted it is slightly more hazardous than lawn bowls.  Protective clothing is part of the reason for that safety.
Beginners must wear long legged trousers (e.g. tracksuit pants), a fencing jacket and a plastron (a half jacket on the sword arm). Girls will also wear a hard molded plastic chest plate. A standard metal mesh helmet with a reinforced "bib" is worn strapped onto the head. A glove is worn on the sword hand.
Sneakers or squash shoes are adequate.
 
The Rules:

There are two fundamental concepts that underpin foil fencing. The first is "target". The second is "right of way".
Target, as the name implies, is that area of a fencer which, if hit by the tip of a foil with pressure the fencer is awarded a point. Valid target in foil fencing is the whole torso, front and back. It is the area covered by a lame. It is not the arms and hands, head, helmet bib, or legs and feet, theses are non valid or off target hits and do not score a point.
All hits both on target and off target stop the fencing bout for the referee to describe the last actions and award points.
A hit to target with the tip of the foil, with force sufficient to slightly bend a steam foil vertically is a valid hit. A flat hit with the side of the blade is no hit.

Right of Way

The rules provide that an attack is made by one fencer on the other, by extending the sword arm so that the foil tip threatens (i.e. points at) target.. That attack either will succeed in achieving a valid hit or it won't. Whichever is the case, the fencer attacking has right of way until the attack hits (on or off target), fails (misses), is abandoned, or is parried.  If it succeeds with a hit the President (referee) calls halt  fencers remain where they are and the president describes the last actions and awards a point if on target or starts the bout where it finished if off target, if a point is awarded the fencers return to the start lines (the on guard line). If both fencers start to attack at the same time and both score a hit then no hits can be awarded and play starts where it finished this is called a simultaneous attack. Right of way transfers to the defender if he makes a successful defense (a parry)

Bouting 

Bouts start with a simple salute and end with shaking of hands with the unarmed hand.

Bouting can be informal and self regulated. Fencers are expected to honestly concede a hit on them to their opponent by indicating if and where a hit has landed on or off target. The bout can continue as long as they wish, scoring or not.
Bouts are usually first to 5 hits or three minutes of actual fencing.  Bouts to 15 hits, with a break every 3 minutes, are used in high level competition.