A lease is a legal agreement by which the owner of a building, a piece of land, or something such as a car allows someone else to use it for a period of time in return for money.
If you lease property or something such as a car from someone or if they lease it to you, they allow you to use it in return for regular payments of money.
If you say that someone or something has been given a new lease of life, you are emphasizing that they are much more lively or successful than they have been in the past.