Go to the Offers tab, tap See details, then Apply. Applying puts you in the running for that delivery job, together with every other driver who applied. Nothing is yours until it's won.
The job's timing decides how far ahead you can apply. A job more than four days out is open to the shipper's preferred and invited drivers, and opens to everyone else as the pickup date gets closer. Before your first delivery job, the app only lets you apply to jobs that start within the next 3 days. Shipper settings can also affect which offers you see and can apply to.
The job goes to the applicant with the highest rating when applying closes, and drivers a shipper marked as preferred come first for that shipper's jobs. Applying the second a job appears doesn't win it, and applying last doesn't lose it. How long a job stays open depends on who's applied and how close the pickup time is: it can close in a couple of minutes, run for half an hour, or close on the spot inside a few hours of the pickup.
If you win, the job moves to your Assigned tab.