In discussions with Arminians, I've often heard election described in words something like this.
When the Bible talks about election, all it means is that in eternity past, God looked down the corridor of time and saw who would believe and who would not. He then elected those He foresaw would believe.
The Arminian view, expressed here in somewhat colloquial language, is called "conditional election." According to this idea, God elects some and not others based on whether or not He foresees the fulfillment of some condition - such as faith, obedience, holiness, etc. It is opposed to the Reformed doctrine of unconditional election. The Westminster Confession of Faith (3.5) expresses the Reformed doctrine in this way:
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 mere free grace and love, 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.
This statement is found within the larger context of a statement on the divine decree, which begins with the words:
God from all eternity did, by the most wise 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.
Although God knows whatsoever may or can come to pass upon all supposed conditions, yet hath He not decreed anything because He foresaw it as future, or as that which would come to pass upon such conditions (WCF, 3. 1-2).
The Canons of Dordt provide a fuller statement of unconditional election. In the First Main Point of Doctrine, Article 10, we read:
The cause of this undeserved election is exclusively the good pleasure of God. This does not involve his choosing certain human qualities or actions from among all those possible as a condition of salvation, but rather involves his adopting certain particular persons from among the common mass of sinners as his own possession.
Arminians reject unconditional election because they believe it destroys human free will. That aspect of the debate has been argued for centuries, but I would like to focus for a moment on a problem inherent in the Arminian doctrine that most Arminians seem to miss.
Arminians do not want to affirm that God has decreed whatsoever comes to pass (i.e. history), and yet Arminians speak of God foreseeing history. Thus the commonly heard phrase, "God looked down the corridor of time." It's another way of saying that God foresees history. Arminians want to maintain a certain concept of human freedom at all costs, and in their view, it can't really be something that falls within God's decree. If it's part of the decree, it isn't really free.
But here's the problem. When the Arminian says that God looks down the corridor of time, what exactly is this "corridor of time" that God sees? If you affirm that God is "looking" at it, you are saying it exists in some sense. But in what sense? The Arminian is saying that God sees all history before it happens, but Arminians also deny the eternal divine decree of all history. So if the "corridor of time" (the events of history) that God looked at did not exist in a divine decree and also did not yet exist in reality, what was God looking at and where did it exist?
For the events of history to exist in the mind of God prior to their existence in external reality would require something like the divine decree, but Arminianism rejects the divine decree. For the events of history to exist "outside" of God prior to their existence in external reality would require either that they both exist and do not exist in the same sense or that they exist in some other mind that God can observe and know. The first option is inherently self-contradictory, and the second option would require something like a second "god" to exist.
One might also suggest that the events of history eternally exist outside the mind of God in reality, but that would force us to deny that God is the only eternal being. None of these are attractive options for the committed Arminian.
In short, if an Arminian is going to talk about God looking at the corridor of time (i.e. history) before those events happen in external reality, he needs to explain the nature of this corridor and where it came from. He rejects the Reformed option, but the remaining options are either self-contradictions or blasphemy.
Perhaps the Arminian should reconsider whether his presupposition about the nature of the human will that created this mess is itself the problem.