@Alex B already answered the first question. I want to respond to the second and third:
I have heard the term "The equity on your home is like a bank". What does that mean? I suppose I could borrow using the equity in my home as collateral?
Yes, you can borrow against the equity in your home. What you should keep in mind is that you can only borrow against the amount that you've paid on your house. For example, if you've paid $100,000 against your house, you can then borrow $100,000 (assuming the value hasn't changed). The argument that this is a good deal misses the obvious alternative: If you didn't spend that $100,000 on a house, then you'd still have it and wouldn't need to take out a loan at all.
Of course, equity still has value, and you should consider it when doing the cost/benefit analysis, but make sure to compare your equity to savings you could have from renting.
Are there any other general benefits that would drive me from paying $800 in rent, to owning a house?
- Your recurring costs (interest payments, insurance, HOA fees if applicable, maintenence) may be cheaper than rent if you're living in an area with high rental costs compared to house prices. (Alternately, those fixed costs may be more expensive if you're living in an area where it's cheap to rent) The calculator @Myles found can help with this, although it makes a lot of assumptions that might not apply to you.
- Your utilities may be cheaper if your new place is more efficient, and if your landlord wasn't paying them. (Or, your utilities might be more expensive, if your new place is larger, less efficient, or if your landlord previously paid them)
- You might be able to save some money by moving less often. (Or you might spend significantly more money moving if you decide you don't like your new house)
- You might end up with a higher net-worth, since the house payments are required and not optional like normal savings. (Although, houses are a poor investment and you might do better with index funds if you have the self-control for it)
As you'll notice from my parenthetical remarks, this is extremely situational. It might be good to come up with a spreadsheet for your situation, taking all of the costs into account, and see if you end up better or worse.
Also, there's nothing wrong with buying a house for non-economic reasons if that's what you want. Just make sure you're aware of the real cost before you do it.