This is pretty basic functionality in Ubuntu, but it's not obvious to new users. To open the Run Application dialog, just hit Alt+F2 on your keyboard. A little dialog will appear where you can type a command. For example, to start Mozilla Firefox, hit Alt+F2 and then type firefox at the prompt and click on Run. The Firefox browser should start up.
For programs that require administrative privileges, you need to enter gksudo prior to the command name. For example, to launch the Synaptic Package Manager, hit Alt+F2 and enter gksudo synaptic and click Run.