If you're irresponsible, you're careless about the consequences of your actions. You can't really rely on irresponsible people.
Being irresponsible is the opposite of being responsible and careful — you do what you like and don't care what happens afterward. Forgetting to feed your dog for a week is irresponsible. Splashing around in a mud puddle in your new white pants is irresponsible. Drug use and unsafe sex are irresponsible. When the government runs up a huge deficit, many people think that's irresponsible. If you have to work with someone irresponsible, good luck: you're going to need it.