What is time to a god?
Part of the problem here is that we mortals, without assistance from the Holy Ghost, really can't comprehend Godliness in any terms but our really narrow field of view.
Now whether there is more than one time appointed for men to rise it mattereth not; for all do not die at once, and this mattereth not; all is as one day with God, and time only is measured unto men. (Alma 40:8, emphasis mine)1
The important part of that statement is that time is important to us, but isn't important (from a perspective that we understand) to God. Before I continue the main theme of what's unique about our belief in eternity, let me bring up one more verse:
Man was also in the beginning with God. Intelligence, or the light of truth, was not created or made, neither indeed can be. All truth is independent in that sphere in which God has placed it, to act for itself, as all intelligence also; otherwise there is no existence. (D&C 93:29-30, emphasis mine)2
When we consider what "eternity" means, it's important to understand that the truth we understand is within the scope of this Plan of God's and not, insofar as we can comprehend, God's time. From the perspective of this Plan of God's, which the Church of Jesus Christ of Latter-day Saints commonly calls the "Plan of Salvation," our Heavenly Father, God, has always existed and always will exist.
Therefore, from the perspective of this "independent sphere" in which God has placed His "truth," the word "eternity" describes the time from the moment our Heavenly Father decided to create that independent sphere until the time that independent sphere's purpose has concluded.
But will God not last forever, and did He not exist into the dawn of all existence?
Here's where things get sticky when you compare our dogma to that of another Christian tradition. Because we believe God had a beginning.
The fifth president of our Church, Lorenzo Snow, is famous for the following couplet:
As man now is, God once was;
as God now is, man may be3
When our Father spoke to Moses He explained:
But only an account of this earth, and the inhabitants thereof, give I unto you. For behold, there are many worlds that have passed away by the word of my power. And there are many that now stand, and innumerable are they unto man; but all things are numbered unto me, for they are mine and I know them.
Our Heavenly Father created a great many worlds and "Plans of Salvation" before this one and has a great many more in-process. And based on what has been revealed, He will continue to create worlds and organize "Plans of Salvation" upon them.
OK, we know from President Snow that our Father had a beginning (I'll get to that in a moment), and we know from Moses (writings restored through Joseph Smith) that our Father has been doing what He's doing for an incomprehensible period of time before Moses.
So, from a human perspective, if we identify "today" as point "zero" on a time graph, then God became God at some time -∞ (negative infinity) and will exist until some time +∞ (positive infinity). That's eternity.
And yet God had a beginning. And that's were we differ majestically from other Christian traditions. We find a curious statement in Psalm 82:6:
I have said, Ye are gods; and all of you are children of the most High.
Now that might cause many to raise an eyebrow, but maybe it can be dismissed by David's artistic license (if David wrote the Psalm). But Jesus referred to the verse:
The Jews answered him, saying, For a good work we stone thee not; but for blasphemy; and because that thou, being a man, makest thyself God.
Jesus answered them, Is it not written in your law, I said, Ye are gods? If he called them gods, unto whom the word of God came, and the scripture cannot be broken… (John 10:33-35)
Jesus agreed that we, the spiritual children of our Heavenly Father, are destined to become gods. I'm sure many other traditions would disagree with that assessment, but it's codified in our dogma:
Wherefore, as it is written, they are gods, even the sons of God—Wherefore, all things are theirs, whether life or death, or things present, or things to come, all are theirs and they are Christ’s, and Christ is God’s. D&C 76:50-70(58-59))
Which reflects President Snow's couplet.
My personal opinion is that it's satisfying to watch science grow ever closer to the truth I hold so dear. The concept of time becomes really messy once you get out of our pleasant solar system — and God knows more about that than we do. He's done His best to give us a reasonable explanation, but not a lot of details, because (when you think about it), how time relates to Him is pretty much irrelevant to our goals as His children here on Earth.
So, despite our belief that God had a beginning, that beginning was further back in time than I can comprehend and He's going to keep on going further into the future than I can comprehend.
That's "eternity," isn't it?
Yeah, yeah, yeah... but something eternity, yeah?
I really can't speak to this. Like @depperm, I don't recognize a something eternity phrase from my experiences. The phrase "contingent eternity" sounds like something a well-meaning but not-too-well-prepared Sunday School teacher might say to describe the differences between the Kingdoms of Glory (c.f., D&C 132:17) or something like that. Inside the Church we distinguish "immortality" and "eternal life" such that "immortality" is granted to all through the Grace of Christ but "eternal life" (aka "exaltation") is only granted to those who are not cast to perdition (and even that's a fairly simplistic statement). I'd vote not to worry to much about what that something was.
1 "God's time" and "man's time" has been a favorite discussion with numerologists. We see, for example, 2 Peter 3:8 and many similar verses that, taken literally, suggest that a day to God is a thousand years to humanity. But the scriptures are full of basic hyperbole (how many times should I forgive? seven-times-seventy... i.e., always). Can you imagine trying to explain to a nine-year-old how something as complex as a computer works? What would you do? Computers affect every aspect of their lives, after all. Of course, what you do is simplify the explanation so that it makes sense to a nine-year-old. What humanity regularly forgets is that, compared to God, we're nine-year-olds. We're old enough to understand the truth... but not necessarily the details. So let's leave you with another passage of LDS scripture, Moses 1:6, "...but there is no God beside me, and all things are present with me, for I know them all." There's only one way for "all things [to be] present with me," and that's for our concept of time to be irrelevant to God.
2 Curiously, science is beginning to understand that the measurement of time between two (or more) worlds is messy. We can identify the relationships between gravity, velocity, and time, but the ability to identify time between two worlds is so complex that Physics has a name for it: the Relativity of Simultaneity. "According to Einstein's special theory of relativity, it is impossible to say in an absolute sense that two distinct events occur at the same time if those events are separated in space." In other words, God actually can't explain the difference in time between His realm and our world without making us a whole lot smarter than we are today.
3 And for the modern viewing audience, "man" is used in the sense of Humanity (aka, "Mankind"). It's a very different world today compared to when I was a child. Back then, everybody understood that the word encompassed everybody. So, please don't get upset — Lorenzo sincerely believed women would become like God, too, and the modern Church believes that anyone, regardless of their gender-related condition, can become like God through obedience to God's commands (D&C 130:20-21).