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Definitely a new word to me..so I looked around google for a better definition

I found it on a Catholic web site ( the dictionaries all define it as eternity )

I think I understand it from this definition

http://www.catholictreasury.info/books/everlasting_life/ev13.php

God has no time line. Humans do because of the world we live in. I think that is the essence of this term.

From the website you linked (Thanks):

Eviternity and Time

What measures the duration of separated souls? [181] We must distinguish three kinds of duration: time, eternity, and an intermediate kind of duration, which is called eviternity.

On earth our duration is measured by continuous time, which is itself the measure of continuous movement, especially of the apparent movement of the sun. It is thus that we distinguish hours, days, years, and centuries. When the soul is separated from the body and is not yet beatified, it has a double kind of duration: eviternity and discontinuous time. Eviternity measures what is immutable in angels and separated souls. It is the measure of their substance, of their natural knowledge of self and God. Eviternity excludes succession. It is a perpetual present. Yet it differs from eternity, because it has had a beginning, and because it is united to discontinuous time which presupposes past and future.

Discontinuous time, then, is opposed to continuous or solar time. It is found in angels and separated souls, as the measure of successive thoughts and affections. One thought lasts for one spiritual instant. The following thought has its own spiritual instant. To illustrate: here on earth a person in ecstasy can remain two solar hours, or many hours, in one sole thought which represents to it one sole spiritual instant. Similarly, history characterizes different centuries, for example, the thirteenth or the seventeenth, by the ideas which predominate in each of these centuries. Thus we speak of the century of St. Louis, of the century of Louis XIV. Hence a spiritual instant, in the lives of angels and separated souls, can last many days, even many years, measured by our solar time, just as a person in ecstasy can remain thirty successive hours absorbed in one single thought.