Defining a variable with or without export

To illustrate what the other answers are saying:

$ foo="Hello, World"
$ echo $foo
Hello, World
$ bar="Goodbye"
$ export foo
$ bash
bash-3.2$ echo $foo
Hello, World
bash-3.2$ echo $bar

bash-3.2$ 

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.