Flat fee arrangement means that a client would pay a flat fee for a certain type of matter. For example, if a client needs a document drafted that they plan on using over and over with their own clients with little expectation of negotiation e.g., a restrictive covenant agreement including confidentiality, non-compete and non-solicitation provisions, employee handbook, marketing agreement, distribution agreement, etc. Flat fees are appropriate here because the amount of time the work could take is predictable.
Compare this to a flat fee arrangement with reduced hourly rate. This is the same as the above, except that there may be negotiations with another attorney or some third party. An example of this may be a commercial lease, whether you are the tenant or landlord, the amount of time necessary to complete the transaction is unpredictable. As this can be more uncertain, the attorney would charge a reduced hourly rate, down from their normal rate, for any hours above an agreed-upon amount of time the initial draft takes, or after an agreed-upon amount of negotiation time is exhausted. Another example of work in this category may be an operating agreement among members of an LLC (like a partnership agreement). If there's only one member, one may not be necessary. However, with other members, including their lawyers, the time could drag on depending on the experience level of the parties and complexity of the issues.
A third type of fee is the straight hourly rate: The lawyer and client agree on an hourly rate and as the work is performed. This merit-based approach tends to be most equitable for both parties in the long run, but not always. For example, in a transactional context, the client may be given a document from a third party and needs the lawyer to review it, e.g., loan and security agreement, promissory note, lease for new retail space, marketing agreement, advertising agreement, etc. In a civil litigation context, the lawyer would draw money from his/her trust account. Usually the lawyer would require the client pay an initial "retainer" amount for a certain amount of hours expected on the matter and replenish the fund as it was reduced. The client should always expect an accounting of time at the end of each month or as the parties agree.
Related to the above fee arrangement, I have, on rare occasion, worked with companies on a deferred billing model...meaning that when funding is secured, usually pretty certain after specific milestones are met, the lawyer gets paid. The lawyer bills at their normal or a little higher hourly rate, which compensates them for the risk of not getting paid at all. This can work so long as the funding is pretty certain and there is a trust relationship established.
A fourth type of arrangement is a straight contingency, from 33 1/3% up to 40%: This is a common arrangement in a personal injury, civil rights, employment discrimination, wage and hour violation suit on behalf of a plaintiff. Lawyers often take these cases on a contingency because the laws on which the suits are based are often "fee shifting" meaning that if the plaintiff is successful, the defendant pays the plaintiff's legal fees. Defendants usually pay their attorneys hourly fees in this scenario. Costs are always deducted from the client's portion of any recovery.
A fifth type of arrangement is a hybrid/conversion: This is used in a classic civil litigation suit where there may be money to recover from a defendant, e.g., for breach of contract. Rather than doing a straight hourly arrangement, a client would pay a lawyer up to a certain amount of hours at a normal hourly billing rate, e.g., 20, and then the relationship would become contingency; say 1/3 for the attorney. If/when money was recovered for plaintiff, the first 20 hours would be subtracted from the 1/3 recovery, preventing the lawyer from getting 1/3 AND his/her initial hourly rate for the first 20 hours, i.e., "double dipping".
A sixth type of arrangement, many companies may chose to hire a lawyer on a retained basis. This works if the attorney has expertise in a number of areas including start-up/corporate/business, real estate and labor/employment. I happen to work with certain companies under this arrangement as it gives the client an opportunity to call me whenever they want without worrying about a clock as well as some certainty when it comes to legal costs and it gives me a steady stream of payment. Often times the client and attorney will agree on a certain amount of hours per month for a certain amount of time devoted to the client. Because the attorney puts aside a set amount of time per month, the money is generally not refundable if not used. However, certain roll-over situations have been done. Finally, certain firms will take "equity" in a client company in exchange for legal services performed. This is very tricky, requires the proper disclosures and is extremely risky from an attorney's point of view. However, it is becoming a more acceptable relationship between client and attorney, if done correctly.