Ideal meaning
An example of ideal is a home with three bedrooms to house a family with two parents and two children.
An example of ideal used as an adjective is the phrase "the ideal job situation" which could be the right blend of hours and wages for a particular person's needs.
The restaurant is considered the ideal in fine dining.
The ideal of national unity.
If (1) the empty set were called a "small" set, and (2) any subset of a "small" set were also a "small" set, and (3) the union of any pair of "small" sets were also a "small" set, then the set of all "small" sets would form an ideal.
Let be the ring of integers and let be its ideal of even integers. Then the quotient ring is a Boolean ring.
The product of two ideals and is an ideal which is a subset of the intersection of and . This should help to understand why maximal ideals are prime ideals. Likewise, the union of and is a subset of .
An ideal work environment.
Ideals are like stars; you will not succeed in touching them with your hands. But like the seafaring man on the desert of waters, you choose them as your guides, and following them you will reach your destiny  Carl Schurz.
Origin of ideal
 From Middle English pertaining to the divine archetypes of things from Late Latin ideālis from Latin idea idea idea
From American Heritage Dictionary of the English Language, 5th Edition

From Wiktionary