1. Placebo effect: Say that a believer wants to do well on a test or run their fastest time in the 100-meter dash. If they pray for that outcome, or know that others are praying similarly, they're likely to do better than they would have otherwise, simply because they believe prayer will work. Even in cases that are beyond conscious control, such as quick recovery from an illness, the mere belief that one will recover quickly—whether due to prayer or a sugar pill—is enough to increase one's immune function and spur a faster recovery. The placebo effect may sound incredible, but it's actually one of the most powerful and well-documented psychological phenomena in existence.
2. Regression fallacy: Everyone knows that life has its ups and downs, but when are believers are most likely to pray? Naturally, they will tend to pray more when things are bad, and to pray the most when things hit rock bottom. And since this point is rock bottom, their situation can only get better from there by definition. But because the believer has just been praying so much, they will often attribute the improvement to prayer, even though things would have gotten better anyway. This idea is so simple that it's easily overlooked, but it explains a great deal.
3. Bandwagon effect: People tend to believe things because other people around them believe those things. Believers often belong to a church and are surrounded by people who believe that prayer works. Thus, they are more likely to believe the same. Even if they begin to have doubts about the efficacy of prayer, seeing the strong belief that others have in it will strengthen their own belief. Note that this self-reinforcing effect allows the belief of the entire group to be sustained, even if the belief of every individual within that group occasionally falters.
4. Wishful thinking: This is another simple yet powerful concept: people are more likely to believe things that they want to be true. The idea that a supreme being can not only hear you anywhere at any time, but can also respond to you and act upon the physical world on your behalf, is incredibly appealing. Conversely, the idea that you are often alone and powerless in the world is highly unpleasant.
5. Confirmation bias: People tend to favor information that supports their existing beliefs. This is a massive factor in understanding believers' perceptions of prayer, and it comes into play in two ways. The first is known as "selective recall." In this case, it means that believers will generally remember answered prayers and forget about unanswered ones. The more unlikely the answered prayer, the more likely it will be to stick in their minds. In this way, instances of allegedly answered prayer seem to occur more often than they really do.
The second point follows from the first: believers will tend to tell other people about answered prayers (and tend not to tell them about unanswered ones). Again, the more improbable the answered prayer, the more likely they are to tell others. Those people then tell other people, who tell others, and so on (and remember, selective recall rears its head at every step). The overall effect is that even though extraordinary examples of "answered" prayer occur only very rarely, they will tend to be heavily reported, so that such examples appear to happen relatively often.
6. Sampling bias: Believers are biased in selecting what they will pray for: they usually pray for things that are likely to happen anyway. They might pray for it not to rain during the few hours that an outdoor party or sporting event takes place. If it doesn't rain, they will interpret this as an answered prayer—even if rain was unlikely during that particular interval. On the other hand, if a family member's legs are amputated following an accident, even fervent believers probably wouldn't pray for those limbs to miraculously grow back. By limiting their prayers in this way, believers tend to get what they expect by purely ordinary means.
7. Postdiction: Among other things, this refers to reinterpreting a prediction after the fact to make it fit with the events that occurred. It's more often associated with alleged prophecies, but it applies to prayer as well. Say a believer goes on a date and prays that they will meet the love of their life, and eventually they end up being very good friends, but nothing more. Although the believer was praying about meeting their future spouse, they may consider their prayer answered—after all, they did end up "loving" this person, but in a different way. Because the criteria that believers use to judge whether they got what they "wanted" are actually much broader than they seem, prayers are more likely to be "answered" purely by chance.
8. The last resort: "God answers all prayers, but sometimes he answers 'no'." This is the one explanation that will never, ever fail. It's completely unfalsifiable—that is, as long as believers put stock in this answer, they will never even consider the possibility that there might not be anyone listening.
The idea is that God only answers prayers that align with his will. There are multiple problems with this. First, it seems to contradict certain verses in the Bible. Matthew 7:7 says, "Ask and it will be given to you." John 14:14 says, "You may ask me for anything in my name, and I will do it." There are no qualifications for these statements, and the context doesn't seem to change the meaning. Believers may argue that Jesus meant to say "I will do it if it's God's will," but that's certainly not what he actually says.
Second, there is the problem of evil issue. Famine, disease, and horrible disasters occur regularly, in spite of prayer and for no apparent reason. Innocent children are forced to become soldiers and sex slaves, again in spite of prayer. Why wouldn't it align with God's will to answer these prayers? In such cases believers can only appeal to omniscience, assuming without evidence that God has some unknown reason for allowing these things to happen.
Finally, If Christian believers still think this last-resort explanation is a good one, I must point out that Muslims, Hindus, and other believers undoubtedly think the same way. Do Christians think these people are justified in thinking that their gods answer prayer in this way, even though this reasoning will probably cause them to continue believing in their false gods? If Christians are justified and other believers aren't, why the double standard? The knee-jerk response is "because we're right," but this is mere assertion—and one that those of other faiths could again use just as easily.
Given the many cognitive biases I have covered here, it's not surprising that believers think prayer is effective. The placebo effect, the regression fallacy, the bandwagon effect, wishful thinking, confirmation bias, selection bias, postdiction, and the ever-lurking last resort explanation all work in concert to form the potent illusion of a supernatural phenomenon. We have many excellent explanations for why prayers may appear to be answered. The onus is on believers to show that they truly are.
Note: It's worth mentioning that we actually have scientific evidence that prayer doesn't work. In the most rigorous experiments on the subject, sick individuals who don't know they are being prayed for (to rule out the placebo effect) fare no better than those who aren't being prayed for at all. Faced with this information, believers now have an even greater hill to climb if they are to show that prayer really is somehow effective.