I think it's past time for another Stupid Questions thread, so here we go.
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Please respect people trying to fix any ignorance they might have, rather than mocking that ignorance.
(I'd be remiss if I didn't link this Mr. Money Mustache post on index funds that explains why they are a good idea)
To buy an index fund, you buy shares of a mutual fund. That mutual fund invests in every stock in the chosen index, balanced based on whatever criteria they choose. Each share of the mutual fund is worth a portion of the underlying investment. At no point do you own separate stocks - you own shares of the fund, instead.
Toy example: You have an index fund that invests in every stock listed on the New York Stock Exchange. The fund invests in $1,000,000 of stock split evenly among every stock on the NYSE, then issues a thousand shares of the fund itself. You buy one share. Your share is worth $1,000. You can sell your shares back to the fund and they will give you $1,000. Over the next year, some stocks go up and some stocks go down. The fund doesn't buy any more stock or sell any more shares. On average, the nominal value of the NYSE will go up by about 7%. The fund now owns $1,070,000 of stocks. Your one share is now worth $1,070.
The dividends go wherever you want them to. The one share of a thousand you bought above entitles you to 1/1000 of the dividends for the underlying stocks in the fund's entire investment. If you're smart, they go to buy more shares of the fund because compound interest will make you rich. You can have them disbursed to you as money you can exchange for good and services, though.
Investing in an index fund is very easy. You will pay by direct withdrawal from a bank account, so you will have to do something to confirm you own the account, but other than that it's like buying anything else online.
Index funds cover costs - which are low, because buying more stock and re-balancing existing stock can be done by a not-that-sophisticated computer program - by charging you a small percentage of your investment. This is reflected by your shares (and dividends) not being worth quite 100% of the fund's value. Index funds are good because they have a very low expense ratio. Many normal mutual funds charge upwards of 1% annually. A good index fund can charge about 0.20%-0.05%. That means you pay your fund about $20 for the privilege of making you about $700, every year.
Opinion time: I own shares in index funds. They are amazing. For a few hours work setting up an automatic transfer and filling out paperwork, I am slowly getting rich. I don't need the money any time this decade, so even if the market crashes tomorrow in a 2008-level event, overall the occasional 1990s-style rises cancel that out, leaving real growth at about 5% assuming you use any dividends to purchase more shares.
I will let you skip the next part of this process and recommend a specific fund: The Vanguard Total Stock Market Index, VTSMX. It invests in every stock listed on the NYSE and NASDAQ. If you have $10k invested in it, the expense ratio is a super-low 0.05, and American stocks are very broad and exposed to world conditions as a whole (this is good - you want to spread out your portfolio as much as possible to reduce risk). Go to vanguard.com , you can figure it out online.
I think I could talk about the minutiae of investing all day. It's fascinating. I should write that post about investing and the Singularity one day.
The key is predicting what will happen to interest rates.