If you don't claim the offer at the time of your first deposit, you can do so at any time within 30 days of registering your account. Simply log in to your account, go to the Account menu, choose My Offers then click Claim Now for the relevant offer.
A bonus bet is basically free swing from a bookie. You place the bet and if it loses you don't lose any money. Bear in mind the Bookie will keep the bonus stake. eg you put a $50 bonus bet on a horse paying $2. The horse wins and the return is $100, the bookie keeps the $50 bonus stake and you keep the other $50.