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Romans 3:28 captures Paul's bottom line well: "a person is justified by faith apart from works prescribed by the law" (NRSV). As a reminder, to be "justified" is to be considered righteous by God, to be found "not guilty" or "innocent" in his divine court. It is to be reckoned as "right" with him. The "works of law" here could in theory be any part of the Jewish Law, but we have followed those who primarily see Paul thinking of those aspects of the Jewish Law that Jews might boast the most about--the ones that distinguished them in their minds from non-Jews. 
But what does Paul mean by "faith." The word faith itself has a number of possible meanings, which we have already seen. It can mean "faithfulness," as in Romans 3:3. It can mean an almost purly mental belief, as when James talks about how the demons "believe" (pisteuo) in one God (Jas. 2:19). Maybe the majority of times in the New Testament it has a sense of confidence or trust, such as when Jesus says we might move mountains if we had enough faith (e.g., Matt. 17:20).
So what sense does Paul have in mind when he says that it is a person's "faith" that justifies them, not their "works of Law"? We are so used to thinking Paul has been talking primarily about faith in Jesus Christ that our knee-jerk reaction is usually to say that it is a trust in Jesus that Paul has in mind here. And faith directed toward Jesus is clearly part of Paul's equation to be sure (e.g., Gal. 2:16). But as Paul unpacks his argument in Romans 4, it is not faith directed toward Jesus that is Paul's focus. It is faith in God the Father. 
So Paul gives Abraham as a model of faith in Romans 4. But it obviously is not Abraham's faith in Christ that is the model but rather Abraham's faith in God. No one earns a right status with God but this status is for the one who "trusts God who justifies the wicked, his faith is credited as righteousness" (4:5). Abraham was considered to be righteous "in the sight of God, in whom he believed [=had faith]--the God who gives life to the dead and calls thigns that are not as though they were" (4:17). And God will justify us too, "who believe in him who raised Jesus our Lord from the dead" (4:24).
As Paul unpacks how faith justifies a person in Romans 4, it is never Christ who is the object of such faith, but God the Father, who raised Jesus from the dead. We see this pattern in Paul's thought also when we look at Romans 10:9, which gives us the path to salvation: "if you confess with your mouth, 'Jesus is Lord,' and believe [=have faith] in your heart that God raised him from the dead, you will be saved." Here again, it is our faith that God raised him from the dead that effects salvation...
 circumcision, purity laws, food laws, sabbath observance, and so forth.
 This is again one of the many observations that I think add up to Romans 3:22 being primarily about Jesus' faithfulness rather than human faith. Jesus is the secondary object of faith for Paul, after God the Father.