So how typically does financing work. Are investors basically giving me a loan that I have to pay back? Are they buying equity (so that upon business failure, nobody owes anybody anything)? Are they doing something in between?
If they are taking equity, how much should I offer them? What are the things I should be considering? I've heard horror stories concerning the dilution of ownership.
I highly recommend reading everything on venture hacks they cover pretty much everything around raising capital.
How much capital do you need to raise? What stage of the business are you at?
I recommend taking a lot of time getting to know your investors and getting to know each other expectations. The first time I raised a seed round of capital the biggest mistake I made was not getting everyone's expectations down on paper, play devils advocate and assume its going to fail and talk through what happens in the different scenarios, discuss what they think they are buying into and what your getting out of them aside from money, what time commitment do you expect from them?
It depends on the investor.
A bank will give you a loan, and expect it to be repaid. In most cases, the bank will ask you to personally guarantee the loan. This means that even if your business fails, you will still have to repay the bank.
A VC, Angel, and most other investors will ask for equity. How much equity you have to give up depends on a lot of factors, including how much money they invest, the current stage of your business, and the type of ownership/stock the investor gets.
If the investor is one of the 3 Fs (friends, family, and fools), it will depend. This gives you the most options, and the most flexible terms, but isn't always an option.
Investors - will expect to be able to mentor you, advise you and put people in the company that can help you move it forward. It's not uncommon for most significant investors to want 51% so they can make controlling decisions and watch out for their investment.