In computers, to execute a program is to run the program in the computer, and, by implication, to start it to run. In usage, people run programs and systems execute them. That is, a system user asks the system to run the program (or sets it up so that this happens automatically at a certain time) and, as a result, the system executes the program. Typically, we don't say that a program is executing; we say that it is running.
An executable is a file that contains a program - that is, a particular kind of file that is capable of being executed or run as a program in the computer. In a Disk Operating System or Windows operating system, an executable file usually has a file name extension of .bat, .com, or .exe. An executable file or a group of them are sometimes referred to as binaries as in "I'll download the binaries to you" since the file format of an executable is a solid sequence of binary values that cannot be easily read by anyone (unlike that of, for example, an ASCII text file which can be easily examined for information as it travels from one computer to another).
A file whose name ends in ".exe" is really a program that when "opened" - that is, selected by putting your mouse over the file name and then initiated by double-clicking your mouse, for example - causes the operating system to run the program. Users who receive an .exe file as an e-mail attachment should always be sure that the file comes from a trusted source and is not, in fact, a computer virus.
Execution is the process of running a program or the carrying out of the operation called for by an instruction.