Ken has provided a very good general explanation. Precise methods for transactions will vary according to state, and even by region with a given state. For example, in some areas, all counter offers are delivered in written form only, while in other areas, counter offers may be communicated verbally following an inital written offer.
Either party may put a time limit on an offer/counter for the length of time the offer will stand. If the contract is not binding upon signature, a time limit is less effective. For example, if there is a provision for an attorney to review the contract before it becomes binding, as exists in some states, the time limit becomes less effective, but might still be ulitlized.
Counter offers can be exchanged several times between the parties before such time that there is a meeting of the minds, or the parties agree to part ways.