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The three things are totally different; there's basically no overlap between them.
**Stock Brokerage: ** Ordinary investors want to buy and sell stock (or commodities, bonds, whatever kinds of investments). Investors don't have direct access to the investment markets, so the broker does the buying and selling for them, and take a commission. Basically a sales agent.
Investment Banking: A company wants to get new money so it can grow. It wants to issue stock in an IPO, or maybe issue bonds so that big investors (pension funds, insurance companies, etc) can lend it some money. The company hires an Investment Bank (or more than one) to sell investors on the idea. The I-Bank sets the price, recruits brokers to sell it to clients, lines up big investors. Also the big investors will hire I-Banks to help them evaluate an investment opportunity -- what SHOULD the price be. I-Banking is a job about information: what's behind the numbers, and where will the numbers be next year?
Hedge Funds: Name is a misnomer -- they're just a kind of private investment fund. A Fund Manager is the company that operates the fund, the fund itself is just a pool of money that investors give the manager, trusting the manager to make good investments. Hedge Funds operate in a world of risk -- basically, they're gambling, but with other people's money, and they're good at counting cards. A lot of hedge funds lose money (at least, relative to an unmanaged index fund), but their investors are really big, and they want to take a risk -- that's why they invest through a hedge fund.
To elaborate on the above comment re: investment banks, one of their most important (and profitable) functions is to guarantee that an offering will raise a certain amount of capital. They do this by giving a firm commitment to the offering (there are other methods to conducting a public offering, but the firm commitment is the most common one, especially for relatively well-established companies).
For example, say Acme Corp. wants to issue stocks to raise $100 million to build a new production plant. Anything less than $100 million will not be sufficient for their needs. They hire an investment bank to advise them on the offering and to set a fair price for the stock, but, no matter how much research they do, they can't be sure of the market's demand for the stock. To guarantee that the offering will meet its goal of raising $100 million, an investment bank (or, generally, a group of investment banks) will agree to buy the stocks being offered by Acme Corp. at a discount from the "fair price." The i-banks then turn around and sell it to the public (or, realistically, to big institutional investors that the banks want to maintain good relationships with). For Acme Corp., it decreases risk and ensures that the offering will raise a certain amount of capital. For the investment banks, they make a big profit from the re-sale of the stock to the public.
Thanks cleared up a few things.
Does a company need to be publically traded to sell bonds to raise money?
No, it's not necessary. It's not unusual for companies that have closely-held (not publicly-traded) stock to issue bonds.
A stock broker does as you say but they also usually offer advice through their direct contact with the individual investor, the Financial Advisor. Now just because that advice often sucks doesn't mean that isn't how the system works... an FA is tasked with asking questions to understand your goals and set up an investment plan that attempts to attain those goals through a reasonable set of investments. They also often tell you when you are straight acting a fool (e.g. your goal is to double your money each year with little risk; just... STFU).
Investment banking includes a lot more than just IPO or raising money through bonds. It includes a wide range of services that center around advice and execution. Some of that is executing transactions: buying another company, selling your company, selling a part of your company, going public, selling additional shares of your company to the market, etc. like those mentioned above. But they also offer a lot of advice AND as an implicit agreement for taking your company public or working with you, they "make a market" in your stock, meaning they actively help find buyers and sellers for your stock, and cover you on the equity research side. The first gives people an easier means of buying or selling your stock, which increases the value of the stock (called the liquidity premium: essentially, if it is easier to sell the stock when I want, I will pay more for it). The second means people understand the stock and company better and when they understand better, they are more willing to invest.
Too lazy to talk about the morass that is Hedge Funds ATM
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