You can "Hide" items bid on in the Bidding section (but that does not remove the actual bid; even if you have been outbid you can still win the auction and if so you are obligated to pay).
You can "Delete" items (at least those which have expired or been rejected, not sure about pending ones and in any case even if you can you should not delete those since it doesn't delete the actual offer) in the Offers section. This used to not be possible (you had to wait until the listing ended--which can be years after your offer--and then delete it from Didn't Win), not sure when they enabled that.
You can "Delete" items in the Didn't Win section.
In each case you check the box to the left of the item you want hidden/deleted (or the box next to the Delete or Hide button to check them all) and then click the Delete or Hide button at the top of that section.
