Re-mortgage actually is moving your mortgage from the existing lender to a new one. In this transaction the security remains the same.
Remortgage is often done when fixed/discounted term on a mortgage expires. The main reason for remortgage usually is to switch to a more attractive rate, but other options are also common like capital raising and debt consolidation.
When consolidating debt into the mortgage it is important to know that the debt becomes secured from unsecured. This means that failure to maintain the payments on previously unsecured debt now means that the lender can repossess the property.