First, it would be helpful to just define the term for the sake of clarity. Omniscience means "all-knowing" and this is derived from "omni" which is a prefix taken from Latin meaning "all", and "scientia", from Latin for "knowledge."
According to many (perhaps most) Reformed theologians, God has foreknowledge of everything NOT because he somehow looks into the future and "sees" what man will do and acts in response; but He Himself has foreknowledge because He has decreed that certain things will come to pass: he does not decree (act in a certain way) because he foreknows, but He foreknows (knows prior to all human action) because he decrees. He knew from the foundation of the world that the elect would be saved and the reprobate would be damned because He decreed that these things would come to pass.
Even though there has been some criticism about a supposed reconciliation between the two, it usually tends to lean toward Arminianism, and so Reformed thinkers have rightly refused any such efforts. One Reformed thinker that holds this view (or some slight variation of it) would be Loraine Boettner:
Foreordination in general cannot rest on foreknowledge; for only that
which is certain can be foreknown, and only that which is
predetermined can be certain. The Almighty and all-sovereign ruler
does not govern Himself on the basis of a foreknowledge of things
which might haply come to pass. Through the scriptures the divine
foreknowledge is ever thought of as dependent on the divine purpose,
and God foreknows only because He has pre-determined. His
foreknowledge is but a transcript of His will as to what shall come to
pass in the future, and the course which the world takes under His
providential control is but the execution of His all-embracing plan. The Reformed Doctrine of Predestination, 99.
Taking the controversial passage in Romans (8:29) that "those whom He foreknew, He predestined..." Calvinists interpret foreknew to be relational, that this indicated a special relationship between God and his elect:
Calvinists contend that the passage teaches that God set His heart upon (i.e., foreknew) certain individual; these He predestined or marked out to be saved. Notice that the text does not say that God knew something about particular individuals (that they would do this or that), but that God knew the individuals themselves: those whom He knew He predestined to be made like Christ. The word "foreknew" as used here is thus understood to be equivalent to "foreloved"--those who were the objects of God's love He marked out for salvation. The Five Points of Calvinism, Steel, Quinn, Thomas. 158
Arminians take it on face value, meaning that even though God is active in influencing them to repentance, that it is a reference to God's knowledge of their future actions, and does not necessarily have any relation to God's action (or lack thereof) in bringing this about. Prominent Arminian theologian says,
...all true Arminians believe in predestination, but not in Calvinist foreordination. That is, they believe that God foreknows every person's ultimate and final decision regarding Jesus Christ, and on that basis God predestines people to salvation or damnation. But Arminians do not believe God predetermines or preselects people for either heaven or hell apart from their free acts of accepting or resisting the grace of God. Arminian Theology, 180.
More could be said, but any further discussion would revolve around specific definitions of omniscience, debates about free-will, debates about future contingents, and attempted reconciliations of divine sovereignty and human freedom and the origin of sin/evil, of which last debate, Calvinists have generally fallen into two camps: the first, that we have no idea how the first sin (whether by Lucifer or Adam) came about. R.C. Sproul held this view:
We sin because we are sinners. We were born with a sin nature. But Adam and Eve were not created fallen. They had no sin nature. They were good creatures with a free will. Yet they chose to sin. Why? I don't know. Nor have I found anyone yet who does know. In spite of this excruciating problem we must still affirm that God is not the author of sin. Chosen by God, 31.
Second is that God did it but this in no way compromises His holiness since there is no standard to which God is subject. Gordon Clark, a famous presuppositional Calvinist apologist held this view:
God's causing a man to sin is not sin. There is no law, superior to God, which forbids him to decree sinful acts. Sin presupposes a law, for sin is lawlessness...the laws that God imposes on men do not apply to the divine nature. Reason, Religion, and Revelation, 240.
In response to the 3 follow up questions about meaning:
God's omniscience is not retrained to temporal thinking: these aren't necessarily technical terms, but conservative Christian thinkers believe that God's knowledge is not limited by free actions (unless, like certain Calvinists, you don't believe in free actions or define freedom compatibilistically), and God's knowledge is not in any way imposed upon by the passage of time. God does not learn anything new or come to believe something that He once believed false, or vice-versa.
God knows everything in the present: this is generally to say that God does not reflect upon any given event as past or future, but is aware of it as a present reality. This is mostly to say that, again, God's omniscience is not restricted by the passage of time. This also is a subject of much debate.
God's choices happen in the eternal-present: this is a rather crude metaphor for God's actions and their performance not being defined by our tensed descriptions of time. To say "eternal-present" is like saying "every moment is now" which is a contradiction, which is why God is often described as transcending time, and so His actions are thus "outside" time.
