Strength, in FFXI, is associated with the element Fire. Valued by melee classes, Strength (STR) boosts Attack Power and Ranged Attack Power at a ratio of 2:1. This means that for every 2 points you have in STR, your character gains 1 Attack Power and Ranged Attack Power. This is different when calculating the Attack Power of a character wielding a two-handed weapon, where the ratio changes to 3 Attack Power for every 4 points of STR.
Strength is also used to determine the damage of attacks against enemies by comparing that enemy's Vitality to the attacker's Strength. The higher an enemy's Vitality is, the lower base damage a character will do against that mob.
All races are not created equally in Vana'diel. Some races have naturally higher Strength stats starting out than others, the breakdown on a level 1 Warrior for all 5 races are as follows:
Possibly the easiest way to boost a stat in FFXI is through consumption of a food. Strength Food comes in a range of prices, the lowest +STR food comes from Hare meat, which allows even the most frugal of adventurers to partake.
You may also choose, once you hit level 75, to put Merit Points into the Strength category to permanently boost this stat.