The "125% loan" is something that is intentional. The lender knows full well that the house is not worth as much as the loan from the begining. Rather rare, but it still worked fine most times since the value of housing was always going up and, an important consideration, the owner got a place to live and a chance of increased equity in the future. Works well if buyer has good income but is short on down payment.

The term "negative equity" means that the loan is larger than the value of the house that secures it. Usually not intentional. Often because housing prices dropped soon after purchase. Also called "upside down loan".

The term "negative amortization" is also confused with the two above. It means that a monthly payment is less than interest so the balance is increased slightly each month. This can lead to "negative equity" situation even in good times.

One more term, "reverse mortgage", refers to a loan that allows an older person who has a house paid off to get monthly cash payments, often until death, and live on the money. Owner who has lived in the same house for years gets to stay but has money to live on. Equity, of course, goes down. This one is a bit tricky as terms are customised for each party.