There are two schools of thought on when the Exodus occurred, the "early" date of 1440BC, and the "later" date of 1290BC. How was each date arrived at, and what is the evidence for each?
The method of deducing each date, along with the difficulties associated with each method, are documented at cresourcei.org
The older date (1440 BC) is primarily based on the assumption that the Bible is a reliable historical document, with the passage of time based solely upon dates and time periods given from within Scripture. Example:
1) 1 Kings 6:1
6:1 In the four hundred eightieth year after the Israelites came out of the land of Egypt, in the fourth year of Solomon's reign over Israel, in the month of Ziv, which is the second month, he began to build the house of the LORD.
This verse gives a time period of 480 years between the exodus and the beginning of Solomon’s work on the Jerusalem temple. From John Bright’s chronology, Solomon ascended the throne around 961 BC, which would make the fourth year of his reign and the beginning of temple construction about 959-957 BC (see Israelite Kings). If we assume that the number 480 is to be taken as a precise number of years much as we would count years on a calendar today, working backward from this date we arrive at a date around 1440 BC for the exodus.
The more recent date (1290 BC) is primarily based on external archaeological and external evidence.
1) Edom and Moab
Archaeology surveys and excavations on the eastern side of the Jordan river (Transjordan), pioneered by N. Gleuck, reveal that there was no settled civilization in the Edomite and Moabite areas of the southern Transjordan until about the late 14th or early 13th century BC. Also, the earliest record referring to the Edomites is an Egyptian letter dating to the 13th century. There is scarcely any evidence of settlement in these areas in the 15th century BC. Since we know from the traditions that Israel encountered settled people in this area (e.g., Num 20:14), it seems that a 13th century date for the exodus is more likely and less problematic than a 15th century date. Also, the Moabite city of Heshbon was the first city taken by the Israelites in the Transjordan area, becoming a part of the tribal territory of Reuben (Num 21:21-24, 32:37). Thorough excavations at what has been identified as this site reveal that the city was not occupied until around 1250 to 1200 BC. Allowing for the 40 years in the desert, this suggests a date for the exodus at the beginning of the 13th century.
These are not the only evidences for each date. It should be noted that there are various assumptions and difficulties with either date. I am not about to list them here, as they are all well documented on the page I linked to above. Suffice it to say that the evidence for each is seen as tenuous, at best by those that hold the opposite belief.
Also, in the sections above, note that I said that each position is based primarily on either internal Biblical evidence or external evidence. That does not mean the position is based only on those two things. Each position is just as much based on an understanding of the difficulties of the opposing view.
Which date we hold to is likely to fall along the lines of how much we trust the Bible for historical accuracy, and how much we trust our interpretation of it, as well as where we draw the line on literalism. (Many of the arguments for a more recent date point out scenarios where a given time span can be interpreted as something other than the direct literal reading. Sounds like another debate about dating certain events that we're all quite familiar with.)