Available on some 'Buy it now' items, there's a feature called 'best offer' giving the buyer a chance to put in an offer for the item being sold.
This is not an auction, so the offer, if accepted is binding and not just a bid. Essentially the best offer system is a good idea, everyone likes a bargain and a chance to save a few quid. A buyer (or potential buyer) can make any offer they like, the seller doesn't have to accept. Do some research on what similar items have sold for and make your offer according to that (For completed items search, click on 'Advanced Search', at the top right of every Ebay page under search box). Don't think because the feature is there that you will be able to buy a brand new XBox for a fiver!
Every seller is different, so what they will accept will vary, I don't think many sellers would accept less than half the asking price, so to offer such would just annoy them. Rather than irk a seller, just ask if they will accept £X on their item, before the offer, they mostly won't bite. If you make an offer and it is rejected you can now make a further offer, I'd recommend contacting the seller if your first offer is rejected. Under the rules, the seller does not have the offerer's id so cannot contact you.
There is also no need to wait with best offers, I would suggest that an earlier offer will stop anyone else from buying it, there's no need for a last minute 'bid', as once the item ends the offer expires and the buyer may decide to sell elsewhere. Quite often a seller cannot respond to an offer made in the last few minutes, so waiting is not a good idea.
Offers last for 48 hours or until item end. If buyer has not replied to the offer, they may be busy or find the offer unacceptable. Its worth remembering that a lot of sellers are professionals and have a living to make, so won't accept every offer made. Offers can also now be automatically declined, so system can be set to reject (or indeed accept) all offers below a certain price. If you offer £1 on something listed at £100, you won't be buying it, offer £90, you've got a good chance.
Any questions, I'll answer them if I can, or ask Ebay, or go to the discussion boards, click on 'Community' at the top of the page. There's usually some good information to be gleaned from the boards (plenty of rubbish too and a fair amount of gossip!).
Remember Ebay's largely a fun and useful environment, just be careful out there!
Recent changes to the system give the seller a chance to put a counter-offer, eg Buy It Now at £30, best offer made for £15, seller can 'meet you half way' with an offer of in this case £25, as I understand you can then reoffer £20. I can't imagine that most sellers will enjoy this, but you never know, it could be fun! Always remember though that the offers are binding if accepted.