What is the mathematical intuition behind àl-jàbrà?

The term algebra comes from the arabic term àl-jàbrà that means "to force", "to restore". Over centuries mathematicians, in east and west, celebrate by this term mathematical disciplines.

What is the mathematical intuition behind this term ("to force", "to restore") and modern derivatives such as in abstract algebra and algebraic geometry?

Append found the following information: In Arabic, al- is the definite article "the." The first noun in the title is jebr "reunion of broken parts," from the verb jabara "to reunite, to consolidate." The second noun is from the verb qabala, with meanings that include "to place in front of, to balance, to oppose, to set equal." Together the two nouns describe some of the manipulations so common in algebra: combining like terms, transposing a term to the opposite side of an equation, setting two quantities equal, etc. Because the original Arabic title was so long, and because it was in Arabic, Europeans soon shortened it. The result was algeber or something phonetically similar, which then took on the meanings of both nouns and eventually acquired its modern sense...

"The word “algebra”—al jebr in Arabic—was first used by Mohammed of Kharizm, who taught mathematics in Baghdad during the ninth century. The word may be roughly translated as “reunion,” and describes his method for collecting the terms of an equation in order to solve it. It is an amusing fact that the word “algebra” was first used in Europe in quite another context. In Spain barbers were called algebristas, or bonesetters (they reunited broken bones), because medieval barbers did bonesetting and bloodletting as a sideline to their usual business. The origin of the word clearly reflects the actual context of algebra at that time, for it was mainly concerned with ways of solving equations. In fact, Omar Khayyam, who is best remembered for his brilliant verses on wine, song, love, and friendship which are collected in the Rubaiyat—but who was also a great mathematician—explicitly defined algebra as the science of solving equations."

(it's from the book - "A Book of Abstact Algebra" by Charles C. Pinter).

From Wikipedia (or rather, its citation of "The Arabic Hegemony" by Carl Boyer):

The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation.

Other uses of the word algebra or its root simply stem from this original use (just like things involving the word geometry no longer actually have to be about measuring the earth).