How does election work according to mainstream Presbyterian thought?

I like how @curiousdannii's answer highlights that Ordo Salutis is logical order, not temporal order (not a sequence in time), and that there is nothing we can do or need to do for the election part of the Ordo Salutis.

You asked:

We cannot offer God ANYTHING But can we not simply respond to Him?

Of course people who are included in election will definitely respond to Him and offer a LOT. In fact, in Reformed theology, it is one of the assurance of election, but it does NOT affect the election part of Ordo Salutis. I think this article by Tim Keller (well published Presbyterian pastor) point #2 answered your question directly:

2. But if everything is fixed and certain, why pray, evangelize, or do anything at all?

This objection is short-sighted. First, if everything was not planned by a holy and loving God, we’d be absolutely terrified by the prospect of even getting up in the morning. Our actions (always done with very little understanding) could have horrible consequences. Everything would depend on us! If everything was not planned by a holy and loving God, there’d be enormous pressure on Christians when we evangelize. We would know our inarticulateness could result in a person missing his or her one “chance” for salvation. It would be a horrible prospect.

Second, we evangelize and pray because of the privilege of sharing in God’s work with him. For example, a father might be able to chop wood for the stove himself, but he asks his children to learn to chop the wood and stoke the fire as well. What if the children say, “We have no incentive to chop the wood. We know that if we don’t cut it, our father will do it anyway—he won’t let us freeze!” But the father would respond, “Of course I could do it myself, but I want you to share the work with me.” The authority and the privilege of working with our heavenly Father is surely plenty of incentive. He wants to work with us and for us.

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Theology has the idea of the Ordo Salutis, or Order of Salvation. A Christian branch like Reformed Theology (of which Presbyterianism is a part) will have a particular way of organising the various doctrines of salvation. The Order of Salvation shows the logical order of these doctrines: how one doctrine leads to and interacts with the others. Note that it is often not a chronological order, and many 'stages' of the Order of Salvation may, from our perspective, happen at the one time.

For Reformed Theology Predestination and Election are the earliest stages and refer to God's sovereign and independent choices. There is nothing that election requires; it is not contingent on anything else. It is purely God's free choice, based on nothing in us.

The Presbyterian doctrine of election is laid out in the Westminster Confession chapter 3.

I. God from all eternity did by the most and holy counsel of his own will, freely and unchangeably ordain whatsoever comes to pass; yet so as thereby neither is God the author of sin; nor is violence offered to the will of the creatures, nor is the liberty or contingency of second causes taken away, but rather established.

II. Although God knows whatsoever may or can come to pass, upon all supposed conditions; yet hath he not decreed any thing because he foresaw it as future, as that which would come to pass, upon such conditions.

III. By the decree of God, for the manifestation of his glory, some men and angels are predestinated unto everlasting life, and others foreordained to everlasting death.

IV. These angels and men, thus predestinated and foreordained, are particularly and unchangeably designed; and their number is so certain and definite that it can not be either increased or diminished.

V. Those of mankind that are predestinated unto life, God, before the foundation of the world was laid, according to his eternal and immutable purpose, and the secret counsel and good pleasure of his will, hath chosen in Christ, unto everlasting glory, out of his free grace and love alone, without any foresight of faith or good works, or perseverance in either of them, or any other thing in the creature, as conditions, or causes moving him thereunto; and all to the praise of his glorious grace.

VI. As God hath appointed the elect unto glory, so hath he, by the eternal and most free purpose of his will, foreordained all the means thereunto. Wherefore they who are elected being fallen in Adam are redeemed by Christ, are effectually called unto faith in Christ by his Spirit working in due season; are justified, adopted, sanctified, and kept by his power through faith unto salvation. Neither are any other redeemed by Christ, effectually called, justified, adopted, sanctified, and saved, but the elect only.

VII. The rest of mankind, God was pleased, according to the unsearchable counsel of his own will, whereby he extendeth or withholdeth mercy as he pleaseth, for the glory of his sovereign power over his creatures, to pass by, and to ordain them to dishonour and wrath for their sin, to the praise of his glorious justice.

VIII. The doctrine of this high mystery of predestination is to be handled with special prudence and care, that men attending to the will of God revealed in his Word, and yielding obedience thereunto, may, from the certainty of their effectual vocation, be assured of their eternal election. So shall this doctrine afford matter of praise, reverence, and admiration of God; and of humility, diligence, and abundant consolation to all that sincerely obey the gospel.