Out-of-order execution (also called dynamic execution) is a technique for improving the performance of a processor by making use of what would otherwise be unused processor cycles. It does this by executing instructions in a different order than they appear in the object code.
A certain amount of 'book-keeping' is required, to make sure that all the operands of a given instruction (especially including those calculated in previous instructions) are available, and that a location is available to store the results - the register renaming mechanism is used to handle that issue.
Out-of-order execution is common in superscalar machines which want to avoid the 'stalling' technique for dealing with conflicts.
It is a valuable technique because it means that software (both programmers and compilers) does not have to worry about execution order, in order to obtain the maximum performance; this is all dealt with by the hardware.