Firstly, not everyone is aware what this is, so on some listings, the bidder is allowed to Buy It Now or alternatively place a 'Best Offer'. This gives the buyer the opportunity to get the item for a slightly lower, haggled price.
This is not like an auction where if you are the top bidder you may still be outbidded, if your offer is accepted it will be as though you will have won it and will be required to pay. Be aware of this and consider your offers carefully! And don't forget to take postage into account.
This is a good chance for the buyer to get a bargain, as they can make any offer they want to, however the seller isn't likely to accept if it is too low.
Now the important bit, do some quick research on what similar or same listings have sold for. You can do this by watching similar items in your My Ebay and seeing what they finish on (remember if they are New or Used they will have different prices, probably). Then make your offer according to what others finished on. Alternatively you can take a short cut and search for completed items by clicking on Advanced Search at the top right of every Ebay page under search box.
As a rule of thumb, no sellers will be ripped off and they will know what the lowest they can go is, it is doubtful that they will accept anything under half price of the buy it now. Because each seller is different and you only have three offers, it is advised to contact the seller before you make an offer. You should ask if they would accept ??? for there item. If they decline and say this is too low, keep a cool head and simply make them another offer via messaging so you are not wasting your 3 precious offers. This is very very useful because the seller does not have the option to contact you unless you contact them first, they will be unable to see your buyer name or identification. If you have already made an offer that was rejected, contact the seller to negotiate and save you BOTH some time.
Make your offers A.S.A.P, before they either get a better offer or decide to hold out and possibly sell elsewhere. If you offer too late and near the end of the listing they might not respond in time and contacting them may prove difficult to say the least!
If the do not reply to your offer, they may be busy or find the offer unacceptable. As a point of principle (being awkward!) they may ignore you if the offer is too low, another incentive to contact first! Some offers may be automatically declined of the seller has set up a system to decline below a certain amount.
For example, on a car worth £4,900. If you offer £100 you will likely be automatically declined or the seller will most certainly decline. However if you offer something in the region of £4,750 you may have a better chance.
Now a seller can make a counter offer, if they want to haggle you up slightly! Mostly though, frome experience and discussions, counter offers are only a small amount below the buy it now price and proffessional sellers are reluctant to budge!
This leads me nicely on to saying that the discussion boards are a great place to ask for help as there are some experienced and knowledgeable ebayers out there!
Good luck :)