The Day Age model is a specific version of the Theistic Evolution idea. Theistic Evolution is an attempt to combine or find compromise between the Creation account found in Genesis with the theory of evolution. The general idea is that God used the methods and means of evolution, such as competition and natural, as the way of creating mankind. The Day Age model states that each of the seven days of creation were long periods of time instead of a literal 24 hour day. These long periods of time are often made to be in the thousands or millions of years.
The passage found in 2 Pet 3:8 is one of the most often quoted passages used in the discussion of the Day Age model. The context of this passage has Peter talking about the scoffers / unbelievers who deny the creation, the flood and the coming of the Messiah. The reasoning used is that since the Messiah has not yet come, he will never come. However, this is faulty logic. Applying this idea of 1000 years is equal to a day to the timing of creation is not justified.
Within Scripture, the topic of creation is always refererred to as happening in a literal seven day period. Any theory that suggests it was a longer period of time has to be based on assumptions, philosophy, or a mixing of evolutionary theory. These produce internal inconsistencies with the creation account. For example, in Genesis 1 how would the plants created on day 3 survive for thousands or millions of years without the sun light created on day 4 and another thousands or millions of years without the insects created on day 5 for pollination? In Exodus 20:8-11 it talks of the Sabbath day rest in regards to creation speaking of six days of work and one day of rest. If the days are interpreted as long ages would it mean, for instance, that man must work 6,000 years and then rest for 1000 years?
One of the prime understandings of Scripture is that when Elohim created mankind there was a special relationship between them. When Adam and Eve were disobedient, this relationship changed and death came into the world (Gen 2:15-17). This leads to the need of having a Savior who can bring people back into that special relationship and the offer of eternal life (Rom 5:17-19). If any type of evolution (gap theory, day age theory, theistic evolution theory, etc.) is used, then the ancestors of Adam and Eve would have been fighting and killing each other for supremecy for millions of years. This would negate the effects of the first sin as death would already exist in the world and a judgment of death for disobedience would be meaningless. Instead of mankind bringing death into the world as stated in Scripture, evolution suggests that death brought man into the world through competition and natural selection.
The scientific evidence for a young earth does not allow for an extended period of time. Examples would include the rotational speed of the earth, concentration of elements in ocean, existence of short term comets, decay of the earth's magenetic field, etc. Such data show a very short time period of existence which is measured in the thousands and not millions of years.