What is the advantage of getting a fixed-rate mortgage?
Because the interest rate does not vary with a fixed-rate mortgage, the advantage of a fixed-rate mortgage is that you always know what your monthly payment is going to be. Thus, budgeting and planning the rest of your personal finances are easier.
What are some of the drawbacks of a fixed-rate mortgage?
If interest rates fall significantly after you have your mortgage, you face the risk of being stranded with your costly mortgage. That could happen if (due to a deterioration in your financial situation or a decline in the value of your property) you don't qualify to refinance (get a new loan to replace the old). Even if you do qualify to refinance, doing so takes time and usually costs money for a new appraisal, loan fees, and title insurance.
If you sell your house before paying off your fixed-rate mortgage, your buyers probably won't be able to assume that mortgage. The ability to pass your loan on to the next buyer (in real estate talk, the next buyer assumes your loan) can be useful if you're forced to sell during a rare period of ultra-high interest rates, such as occurred in the early 1980s.
Fixed-rate mortgages sometimes have prepayment penalties.