Here are the verses in question from the NASB:
Now it came about after these things, that God tested (KJV - "did tempt") Abraham, and said to him, "Abraham!" And he said, "Here I am."
Let no one say when he is tempted, "I am being tempted by God"; for God cannot be tempted by evil, and He Himself does not tempt anyone.
The words that have been translated "tempt" in these two passages come from the Hebrew (in Genesis) and the Greek (in James). They are different words, from different languages, but they do carry much the same meaning. In brief, they can mean any of the following:
In order to determine the specific meaning of the words in the particular places where they are used, we need to look at context.
The passage in James, contextually, is addressing people who are claiming that God is trying with all His might to get them to do something sinful. Their argument may have been that if they succeed in not giving in to the temptation, they have conquered God, and if they fail and succumb to the temptation, it is not really their fault. James is chastising these people by telling them how idiotic they are being. God doesn't do that.
In the passage in Genesis, God is clearly not Doing His Best To Get Abraham To Do Something Sinful, but rather, has given Abraham a command that Abraham is to obey. Here, Abraham was proven faithful by his obedience to God's command, as Abraham in fact was going to obey it (until he was prevented from doing so by the intervening angel).
So the contexts, and therefore the intended meanings of the specific words, are totally different in these two passages.
1. God does not try with all His might to get people to do sinful things (James).
2. God does provide people with tests to prove them faithful (Genesis).
If God is the Cause of everything that has ever happened, then someone may ask how we can claim that He does not tempt people, since we see that in fact people are tempted.
Has not God caused this?
Yes, He certainly has.
However, with God, in the words of the great sage, Yoda, "Do, or do not. There is no 'try'." God doesn't "try" to do things. He either does them or He does not do them. He may cause Satan to tempt, and He may cause us to be tempted, but He Himself does not tempt. It is inconceivable for God to be "trying" to do anything at all, since "He does whatever He pleases" (Psalm 115:3). Since He does not "try" to do things, He does not "try" to get us to sin. In other words, He Himself does not tempt us.
Someone will say that it is the same thing. If God causes Satan to tempt us, then it is really God who tempts us. They may say that a nail was driven in by a carpenter, even though the nail was actually driven in by the hammer, which was caused to do the driving by the carpenter. They will say that it is not wrong to claim that the carpenter drove in the nail, and likewise it is not wrong to claim that God tempted if He causes Satan to tempt us. But this is a false analogy.
A better analogy would be to compare it to the scene in "Back to the Future III" where Marty McFly meets Mad Dog Tannen in a bar, Mad Dog gets mad, shoots at McFly's feet and yells "Dance!" McFly proceeds to dance back and forth, even venturing into a Moonwalk. Now here, Mad Dog caused Marty to dance. But note that Mad Dog himself didn't dance.
Likewise, God causes the devil to tempt (2 Samuel 24:1 and 1 Chronicles 21:1; Job 1:21; 1 Kings 22:23), but God Himself doesn't tempt (James 1:13).
The sovereignty of God can be a very difficult issue for many people, particularly in Western culture, to understand and to accept. If you ever have the time and inclination, I'd highly recommend Martin Luther's book, "The Bondage of the Will".