Applying puts you in the auction for that delivery. The job leaves your Available tab while the auction runs and sits under Applied, which is why it can look like it vanished right after you applied.
The auction stays open for a few minutes and then compares everyone who applied. Ratings are what it compares: the ratings your deliveries earn, and the rating a business gave you if you have worked with them before. Drivers a business marked as preferred come first for that business's deliveries. Applying the second a job appears does not win it, and applying last does not lose it.
How long it stays open depends on who is in it and how close the pickup is. It can close in a couple of minutes or run for half an hour, and within a few hours of pickup it resolves on the spot. If you win, the delivery moves to your Assigned tab and a notification arrives. If someone else wins, it leaves your Applied tab.
An application that has been sitting for a couple of hours does not win on its own. A push arrives reading Confirm to take a job you applied to, and opening it and tapping Accept puts you back in. Until someone answers that, the delivery stays open.