When you do something desperately you do it with extreme urgency. If faced with a life-and-death situation, you might fight desperately, using any means possible to overcome the obstacles and survive.
The undercurrent of the adverb desperately is one of despair, which makes sense when you consider that it comes from the Latin desperatus, meaning "given up or despaired of." You can get a good sense of the word from its context in Journalist Sheilah Graham’s statement, "You can have anything you want if you want it desperately enough. You must want it with an inner exuberance that erupts through the skin and joins the energy that created the world."