Vincent Gable’s Blog

on ApplicationIsRunning(appName)
	tell application "System Events" to set appNameIsRunning to exists (processes where name is appName)
	return appNameIsRunning
end ApplicationIsRunning

Use it like,

if ApplicationIsRunning("Mail") then
	display dialog "Don't forget to write mom!"
end if

On Mac OS X 10.5, this worked for me even when the application-file’s name was changed. On 10.4 I do not expect that it would still work if someone renamed the application, unless you used the creator type to locate the process, not the name.

You might also be interested in how to get the name of the frontmost application, here’s how:

tell application "System Events" to set FrontAppName to name of first process where frontmost is true

if FrontAppName is "DVD Player" then
	display dialog "Get to work!"
end if