equity = value of house - amount still owed
most ppl use the word mortgage to define their monthly payment, ie "How much is your mortgage?", but technically a mortgage is "a legal document used as security for a debt." that just means that when you borrow money from a lender (a bank) to purchase a house (or borrow money against the equity) the lender gets the house in case of a default. but i think you mean mortgage in the sense of a home equity loan, which is simply borrowing money against the equity in your property. for example lets say john and marry purchase a house with an FHA loan (little to no down payment) for 100k and 4 years later their house has gone up in value to 120k. after 4 years they now owe 95k on their house. John and marry have 25k in equity. john and marry might be able to borrow up to 25k depending on their credit rating, debt-to-income ratio, and a few other factors.
What if you have your house totally paid for? Do some people still take out a loan against it?
this is very common. a lot of people use the equity in their property to pay off high interest debts like credit cards, and others use it to do renovations on the property itself.
In a nutshell, can anyone explain the basics behind the various "options" people have with houses and so forth?
this is too vague to answer. do you want to know who should/can buy/sell? who should/can borrow?