As stated on eBay's Making a Best Offer help page:
❝You can make a Best Offer on items where you see 'Make offer' below the Buy It Now price. The seller can decide whether to accept, reject or counter your offer.❞
The seller is not obliged to accept your offer. There is no contract where there is no agreement (consisting of offer and acceptance of the offer), consideration (usually money in exchange for goods/services), capacity (i.e., not a minor, and of sound mind), intention (it has to be clear at the time that the parties are entering into a binding and legally enforceable contract), and certainty (the contract has to be certain, complete, and unambiguous with regard to both parties' rights and obligations).
The process and status of making an offer on eBay (where Best Offer/Make Offer is enabled on a listing) goes like this:
Buyer makes offer ↠ Seller has not responded to the offer ↠ no contract is entered into. While the offer remains unaccepted by the seller, the status can be considered "in limbo". (The seller does not have to explicitly reject the offer. Note also that by making the offer, the buyer is agreeing in advance to a contract if the seller accepts the offer. eBay explains that a Best Offer is "binding", but they mean that the Best Offer is binding upon the buyer, not the seller. The seller doesn't have to accept any Best Offer.)
Buyer makes offer ↠ Seller has not yet responded to the offer ↠ buyer withdraws offer ↠ no contract has been entered into, and the seller cannot accept the offer. (However, if the seller has already accepted the offer, the buyer cannot withdraw it.)
Buyer makes offer ↠ Seller rejects offer ↠ no contract is entered into. The status is no longer "in limbo"; it's now explicitly rejected. There's nothing binding upon either the buyer or the seller.
Buyer makes offer ↠ Seller makes counter-offer ↠ no contract is entered into. The seller has implicitly rejected the buyer's offer by making a counter-offer, and the status is "in limbo" with the buyer being the one who can end the limbo by either accepting or rejecting the seller's offer. (That is, by making a counter-offer, the seller is agreeing in advance to a contract if the buyer accepts, just as by making an offer, the buyer is agreeing in advance to a contract if the seller accepts. eBay explains that a counter-offer is binding upon the seller if accepted by the buyer. Counter-offer (just like buyer's offer) remains active for 48 hours; once it has lapsed, the offer is no longer binding upon the seller.)
Buyer makes offer ↠ Seller makes counter-offer ↠ Buyer accepts counter-offer ↠ a contract has been entered into. By the seller making an offer which the buyer accepts, agreement has been created, and the buyer is expected to pay under the terms of consideration, and the seller is expected to supply the goods under the terms of consideration.
Buyer makes offer ↠ Seller makes counter-offer ↠ Buyer rejects counter-offer ↠ no contract is entered into.
Buyer makes offer ↠ Seller makes counter-offer ↠ Buyer makes counter-offer to the counter-offer ↠ no contract is entered into. The status is "in limbo". If the seller rejects the counter-offer, there is nothing binding upon either buyer or seller and no contract is entered into. If the seller accepts the counter-offer, that offer is binding upon both parties. If the seller neither accepts nor rejects the offer, it becomes inactive after 48 hours and is no longer binding upon the buyer (or upon the seller) and no contract is entered into.
... and so on.
Specifically in your situation...
Buyer 1 makes offer ↠ Seller doesn't respond to offer ↠ no contract has been entered into with Buyer 1
Buyer 2 makes offer ↠ Seller responds to Buyer 2's offer with acceptance ↠ contract has been entered into with Buyer 2
If the seller chooses to accept Buyer 2's offer, it's irrelevant as to whether your offer is higher or lower than another buyer's offer. Making an offer is not the same as placing a bid on an auction item. On eBay, the highest bid on an auction item is the winning bid... but with making an offer, the seller is able to consider any offers and make a decision based on any factors that he/she likes. The seller doesn't have to tell you that your offer is rejected; your offer automatically becomes inactive if the seller accepts another buyer's offer.
Buyers should be aware that the only way in which to be certain that they've secured an item listed as BIN with Make Offer enabled is to BUY the item rather than make an offer. Making an offer always carries with it the risk of the offer being rejected, or timing out, or with another buyer's offer being deemed more acceptable by the seller, etc.
