Almost anyone can pick up a weapon and fight to some degree, but few possess the skill and ability to truly call themselves a fighter. Those with the fighter class are usually professional warriors trained as heavy (i.e. armored) infantry or cavalry. In practice, they tend to be knights, soldiers, mercenaries, or sometimes Gladiators, though outlaws and thugs could also be fighters.
Being a professional man at arms puts you in a higher social class than most folk (even if just the fact you can take things by force if you want), and fighting is the traditional duty & privilege of the noble class. In most civilized areas, commoners and tradesmen are restricted from going armed or armored, unless conscripted or in the service of a lord.
A knight is trained in fighting and horsemanship from a young age. He apprentices as a squire to an experienced knight until he is deemed fully ready, and able to train squires of his own. He must be knighted by a lord, to whom he swears allegiance, along with vows to uphold justice, fight honorably, etc., which vary between regions and knightly orders. A knight who looses his liege and cannot find another to take him on, looses honor and will often become a mercenary or brigand.
Knights must provide their own armor, weapons, horses, and retainers, which can get quite expensive. They are bound to serve and fight for their liege, who in return is obligated to support his knights either through a salary and gifts, or by granting land.
In tribal societies, all adult men are expected to be warriors (as well as hunters or herdsmen or whatever). In game terms, they are more likely to have the ranger or barbarian class, as befits their lightly-armored, skirmishing style of fighting.
Women, being on average smaller and weaker than men and constrained by motherhood and traditional roles, are not commonly found in the profession of arms. That is not to say that a woman cannot become a knight, warrior, gladiator, or brigand, but she will face prejudice from many males, and is quite likely to be underestimated, too.
See Also: Warfare, Class