A mortgage is a long-term loan with a structured repayment schedule that enables you to buy your own home. A remortgage allows you to move your mortgage to another lender, or possibly negotiate a new deal with your own, while remaining in the property.
The reasons for doing this are quite varied but most people's prime motivation is the fact that remortgaging your home loan can save you quite a lot of money. This is especially the case at a time when interest rates are low. Most mortgage lenders will transfer you to their Standard Variable Rate at the end of your fixed or discounted deal and switching lenders can pay off in the long run.
You can try to remortgage through your current lender, although it's worth remembering that mortgage lenders tend to reserve their best deals for new customers so you need to shop around.