A mutual fund is what you get when an investment company packages up a bunch of different stocks and/or bonds and sells shares of this package to their customers.
Mutual funds can basically fall into one of two categories: actively managed funds or index funds.
Actively managed funds employ a person (or group of people) to make decisions about what stocks to buy, based on their opinions of which ones will do the best in the market.
Index funds just buy shares according to a formula, tracking a particular index.
The S&P 500 is one index, a list of 500 large US companies. Dozens of other indexes exist, for smaller companies, mid-sized companies, all companies put together, international companies, real estate investment companies, and many more. Vanguard has funds that track many of these indexes.
Because index funds invest according to a set formula, they don't need to pay someone to guess which stocks will do better than others, and they often have lower fees as a result.
VBINX is a mix of 60% CRSP US Total Market Index (which represents all US-traded stock weighted by company size) and 40% Barclays U.S. Aggregate Float Adjusted Bond Index (representing a wide variety of US bonds). VBINX isn't a one-size-fits-all "best" fund, but if you think a 60/40 stock/bond ratio makes sense for you, this could be a good fund to invest in.
You're generally right about the mechanics of it.
1) Whenever you have some money to invest, you'll buy more shares of your fund. Each share represents a piece of the overall fund that owns thousands of different stocks and bonds.
2) Every day, the price of a share is reset to reflect the new value of everything the fund holds, based on what happened in the markets that day. When the market goes up, the value of your shares will go up. When the market goes down, the value of your shares will go down.
3) The fund will pay out dividends periodically, based on how much the underlying shares produce in dividends and interest. You can check Google Finance or other sites to see how much the dividends have been in the past. Often you can set up your account so that dividends are automatically reinvested into new shares.