This analogy is strained, but it may help you to get the gist of it. Imagine you have a bucket that can only hold 10 gallons of water. If you try to put 15 gallons of water in it, it will overflow, spilling everywhere. By definition, 15 gallons of water cannot fit into 10 gallons of bucket. The key difference is that, in software, if you do not account for the possibility of more data being poured into a container than the container can hold, the data can "spill" over the edge of your allocated space, possibly overwriting something else, perhaps code, or other data.
Maybe this will help?
In exhibit A, each of those 16 squares represents a location in memory (naturally there is thousands of time more memory than this, in a computer). We've set aside the first ten (0 to 9) cells for our own purposes (filled with zeros, denoted by A.1). This area of memory, at the moment, has no meaningful data in it. We set it aside, specifically, to store data from the user. In computerspeak, this is called a "buffer".
Notice that my name (well, the first six letters) are DIRECTLY after the space we've set aside. If we look in exhibit B, we've filled out space with five characters (B.1). We still have some left over, so this isn't a problem. My name is still intact and everything is peachy. We used less space than we thought we would need. It doesn't hurt to overestimate.
Now look at exhibit C. This is an example of a "buffer overflow". More data has been written than we aside space for (C.1). Notice that my name (C.2) is no longer intact. We've overwritten it with the overflow data from the Gettysburg Address. This happened because, in our program, we didn't check if we set aside enough space before trying to fill the buffer with the Gettysburg Address. Strictly speaking, there was enough room in memory for the Gettysburg Address (It's trailed off the side of the page), but we accidentally clobbered what was there, to begin with. Ideally, we would have truncated the Gettysburg Address to the first ten characters ("FOUR SCORE") or thrown an error.
EDIT: If you're still curious, PM me.