Defining a variable with or without export

export makes the variable available to sub-processes.

That is,

export name=value

means that the variable name is available to any process you run from that shell process. If you want a process to make use of this variable, use export, and run the process from that shell.

name=value

means the variable scope is restricted to the shell, and is not available to any other process. You would use this for (say) loop variables, temporary variables etc.

It's important to note that exporting a variable doesn't make it available to parent processes. That is, specifying and exporting a variable in a spawned process doesn't make it available in the process that launched it.

To illustrate what the other answers are saying:

$ foo="Hello, World"$ echo $fooHello, World$ bar="Goodbye"$ export foo$ bashbash-3.2$ echo $fooHello, Worldbash-3.2$ echo $barbash-3.2$ 

Others have answered that export makes the variable available to subshells, and that is correct but merely a side effect. When you export a variable, it puts that variable in the environment of the current shell (ie the shell calls putenv(3) or setenv(3)).
The environment of a process is inherited across exec, making the variable visible in subshells.

Edit (with 5 year's perspective): this is a silly answer. The purpose of 'export' is to make variables "be in the environment of subsequently executed commands", whether those commands be subshells or subprocesses. A naive implementation would be to simply put the variable in the environment of the shell, but this would make it impossible to implement export -p.