I'm not 100% on the details, but here goes -
LOI is normally the fist stage in buying aircraft. The airline signs a LOI with the manufacturer which is effectively a contract saying "We intend to purchase N of your XY123 model." So the airline has basically committed to buying the planes, but the exact details haven't yet been worked out. I don't know if a LOI costs an airline if they later don't purchase the aircraft. While this may be announced to the world, sometimes it's just kept under wraps for business reasons. The manufacturer may allocate production slots to the airline based on the details of the LOI.
A firm order is basically the final thrashed out contract that says "We are buying 6 XY123-100s and 4 XY123-200s" The price will be fixed, as will exact delivery dates. This is normally when the manufacturer officially adds the order to their books. This why you often see two press releases - the first saying airline X has selected aircraft Y for its fleet. The second is then confirmation that the firm order has been signed.
At the same time as placing the firm order, the airline may take some options. These are normally price protected within a certain time limit. So, for example, the airline might be able to purchase an addition 10 aircraft (assuming that's how many options they took) at the same price as the ones they just firm ordered, but they have to convert the options into firm orders within a fixed time frame (e.g. two years).
As with any big business transactions, there will be clauses and conditions written in. Also, depending on how big the customer is, the manufacture may be prepared to extend the time limit for the expiration of the options. I don't believe there's any financial penalty for an airline that doesn't convert options to firm orders. Neither Airbus nor Boeing lists options on their website, although they normally announce how many options the airline took. For example, Spirit signed a firm order with Airbus for 15 aircraft, and options on 50, this week.
Time flies like an arrow. Fruit flies like a banana!