My answer will not necessarily be concerned with the scientific study of dreams, but will focus on the Biblical foundation for the definition of a dream, the cause of a dream, the reality of dreams, some common purposes for dreams and the connection or lack of connection between dreams in the Bible and dreams today.
The Definition of a Dream
Once again I want to emphasize that we will define a dream using the scriptural definition. We often speak of day dreaming or following our dreams in addition to our standard concept of dreaming, but here we will focus on the dreams that we have during sleep. The Bible defines a dream in Job 33:15, "In a dream, in a vision of the night, when deep sleep falleth upon men, in slumberings upon the bed;". The Bible says several things in this one verse. First of all, a dream is a vision of the night. The verse also teaches that a dream occurs when deep sleep falls upon men. According to the Bible, a vision is something that occurs when a man is awake, while a dream is something that occurs when a man is asleep. Some who study dreams say that we are always dreaming when we sleep, but we only remember the dreams right before we awake. This would actually fit with the scriptures in several cases where men remember a dream that actually woke them up.
The Cause of a Dream
From everything that I have read about dreams, it seems as though we still do not have solid answers as to what causes them. The Bible, however, does contain a verse that quite possibly gives us the cause. Consider Ecclesiastes 5:3, "For a dream cometh through the multitude of business; and a fool's voice is known by multitude of words." Now it is interesting that several who study dreams suggest that when your body slows down to rest, your mind continues to work. The Bible says that the multitude of business will cause dreams. This could be the multitude of physical and emotional business or it could be the multitude of mental business, but either way it appears that this is what causes us to dream. Though your body settles in rest, your mind can keep right on working and thinking.
The Reality of Dreams
Dreams can appear to be very real. Sometimes you wake up scared, sometimes mad and sometimes sad, yet you realize after a while that it was just a dream and you settle down. The Bible speaks of this in Isaiah 29:8 when it says, "It shall even be as when an hungry man dreameth, and, behold, he eateth; but he awaketh, and his soul is empty: or as when a thirsty man dreameth, and, behold, he drinketh; but he awaketh, and, behold, he is faint, and his soul hath appetite". You may dream of eating or drinking, but when you awake you are still empty. It may have appeared real, but it was not.
Something else to consider here is that dreams are not necessarily supposed to mean anything.
Consider Ecclesiastes 5:7:
"For in the multitude of dreams and many words there are also divers vanities: but fear thou God."
This verse tells us that a multitude of dreams can be worth absolutely nothing. We are not to trust in them, however real they seem; we are rather to fear God.
Common Purposes for Dreams
In the Bible, especially the Old Testament, the Lord used dreams for various reasons. We will not state all of them, but will give you a few to consider.
There were times when God used a dream to give a warning. Consider the twelfth verse of Matthew 2 where the Bible says, "And being warned of God in a dream that they should not return to Herod, they departed into their own country another way." The Lord appeared to the wise men here for the purpose of warning them not to return to Herod, but to go back home another way. In the very next verse Joseph is also warned of the Lord in a dream to flee into Egypt with the Lord Jesus. The very first dream that I have seen in the scripture is when God spoke to Abimelech and warned him that he had taken another man's wife. Obviously this is a common purpose of dreams within the context of scripture.
There were times when God used a dream to reveal prophecy. Joseph was a man subject to dreams and the interpretation of dreams. In Genesis 37, Joseph dreamed a dream about his brethren bowing down to him. If you know the story of Joseph, you know that the dreams are the only evidence of this occuring until he was promoted to the second in command under Pharaoh. For approximately 13 years he had nothing else but these dreams that told of the future. If you are familiar with the Book of Daniel, you will remember that Nebuchadnezzar dreamed dreams that have a tremendous amount of prophecy in them. There are other dreams in the Bible that I would consider to fall into this category as well.
There were times when God used a dream to encourage. I would have to say that this is nowhere near as common as the first two, but I do believe there is an instance that could be construed as a dream for the purpose of encouraging obedience to the Lord. In Judges chapter 7, we find the Lord thinning out the army of Gideon before he goes to battle with the vast army of Midian. Gideon begins with 32,000 soldiers. In verse 3, we find that 22,000 went home because of fear and in verse 6, we find that 9,700 went home because of the way they drank water. This left Gideon with an army of only 300 soldiers. In this same chapter the Lord instructed Gideon that if he was afraid he should go down unto the host and then he would be strengthened. When Gideon arrived in the host, he heard a man telling a dream to another man. The interpretation of the dream just so happened to be that the army of Gideon was going to defeat the Midianites. Immediately upon hearing of this dream, Gideon was strengthened and he worshipped God. Surely the whole purpose for this man's dream was the encouragement of Gideon.
Dreams of the Bible and Dreams Today
Now here is where people will begin to split from me and call me a heretic and all sorts of other things. I personally believe that God does not speak to men through dreams and visions today as He did in the Bible. Consider some things with me. First of all, God used dreams in the Bible to warn men of sin and wrong, but today we have His completed word that will keep us from sin. As a matter of fact the Psalmist said, "Thy word have I hid in mine heart, that I might not sin against thee." Something else to consider is that dreams were often for the purpose of revealing prophecy, yet we have in the word of God the revelation of the future events that God wants to make known to men. Another point to consider is Joel's reference to prophesy, dreams and visions. In Joel 2:28 he says, "And it shall come to pass afterward, that I will pour out my spirit upon all flesh; and your sons and your daughters shall prophesy, your old men shall dream dreams, your young men shall see visions:". Now this verse is quoted as coming to pass in Acts 2:17, yet any honest student of the Bible knows that 1.) not all of what Joel said was fulfilled on Pentecost and 2.) the signs and wonders that were associated with the age of the apostles were for the Jews and went out of business during the ministry of the apostle Paul. Yet Joel still prophesied that dreams will again be a means whereby God speaks to men, but in the context of Joel 2 it is easily understood that this will occur during the time of the tribulation.
Now, let me clarify something on the other end of this argument. I very strongly believe that God can and will allow people to have dreams that could turn them to Him. For example, I believe God would allow a man to have a dream about hell, if it would keep him from going there. I just don't believe that we can give our dreams the weight that we can give to Bible dreams. Ultimately when we dream and it causes us concern, we ought to go to God and ask Him if He is trying to teach us something. If the Lord does not burn it into your heart that He is trying to teach you something then chalk it up to a multitude of business and go on about your day.