Within the United States, most airlines allow you to check luggage weighing up to 50 lbs. for the fee stipulated on the airline website.
Airlines based in the U.S. usually stick to the 50 lbs. per bag weight limit for economy passengers on international flights, but first-class and business-class passengers on both United and Continental are allowed to check bags weighing up to 70 lbs.
Airlines around the world have different policies, but it is common to see international airlines allow a standard weight for all combined checked luggage, regardless of the number of pieces. For coach, this can often be limited to 44 lbs., while business-class passengers are allowed to check 66 lbs. of luggage and 88 lbs. are allowed to the first-class passenger.