Short-term mortgages offer lower interest rates than long-term mortgages. You save money by the lower interest rates and shorter payment period. The trade off is a larger monthly payment, but this option can save you thousands.
If you are making payments on a long term loan, say, 30 year mortgage for the past 10 to 20 years, then refinancing to another 30 year loan will not be a good option as it may increase your overall payment.
Pull your credit report from the bureaus and review it for any negative items (late pays, collections etc) and inaccurate detail. Try to dispute negative items and remove them from the report. If required pay off any unpaid debt. Otherwise, you won't get a low rate and may not even qualify. Of course there are lenders in the subprime market who may offer you a bad credit refinance loan, but it's better to avoid them as they'll possible charge higher rates and fees.
If you plan on moving out of your existing home within the next few years, it may not be beneficial for you to refinance. Make sure you let your mortgage specialist know your future plans.