Integer overflow refers to the phenomenon that occurs in certain computer data types where their signs switch from positive to negative, or vice versa, when they reach the ends of their applicable ranges. In computer ranges, integer data types have circular ranges, and when they reach one end of their range, they immediately move to the other end of their range. This is also called integer overflow.
A signed integer can hold a range of values from -231 to (231) - 1. This integer cannot have a value of (-231) - 1; rather, the next number it increments to is at the other end of its range: (231 - 1). The change from negative to positive at the end of its range is an example of integer overflow. By the same token, an integer cannot have a value of 231; this value would instead switch to the other end of its range and become -231.
This overflow has significant consequences while programming. An array can only have as many indices in it as the integer type allows, and negative indices don't count. If a programmer attempts to create an array larger than the integer type allows, significant memory errors can occur because integer overflow would result in a negative index. This is especially dangerous in languages that don't have explicit bounds checking for arrays, like C++.
When integer overflow occurs, related types of overflow such as buffer overflow, heap overflow, and stack buffer overflow can occur. In all of these cases, the integer overflow acts to overwhelm memory structures with more data than those structures can conceivably hold. These overflows, in simple programs, don't often do much more than cause an invalid read or invalid write error. Manipulation of this problem by hackers, however, can engineer memory errors that can cause more serious problems.
In most simple programs, integer overflow is not a major problem. The limits of the integer type are sufficiently large enough that the overflow problem doesn't come into play unless a lot of data is being handled at once. In some cases overflow can be mitigated, as in the case of incrementing counters, by using a larger data type with a greater range. The larger data type could, in theory, eventually encounter the same overflow problem, but as data type ranges get larger, the chances of doing so get smaller. Each integer data type's range is at least twice the size of the next smallest, so there is ample room for additional data.