A person who invests in very early stage companies in return for equity. Often, but not always, a former entrepreneur who has made a good deal of money. See Google.
If you want a more comprehensive overview of the different archetypes of investors, then Guy Kawasaki's The Art of The Start has a good overview, and is a fun read.
From your point of view, a business angel is a person who has something you need to progress with your venture, and is willing to provide it, in exchange for a share in your business.
In almost all cases, that something shouldn't just be cash. Sure you need cash, but you also need reputation, contacts, capability, experience, someone invested in your success who can help you with big decisions like hiring.
If you are offered a big slice of that - and you can satisfy yourself it's all real - then you've found someone who can be a business angel for you.
The caution you'll hear is that a whole lot of people who describe themselves in this way really aren't going to be an angel for you. So if you're new to the game, confine your attention on people you can check
You may be able to shortcut a lot of this if there are functioning angel networks in your area. If there are, research them, their membership, their interest areas and anything they have that gives you an idea how closely they operate. Then find out how they facilitate people pitching.
No decision about equity is risk-free. But the risk you're taking with your founding team and early hires is usually far, far greater than the risk (when you've managed it) with your early investors.