I see everyone saying "index", but there are tons of index funds to choose from. I need to understand what makes them different.
The difference is (mostly) the index the fund is based upon. Indices are groups of stocks sharing some similarity that people track to see how the stock market is doing. The big one in index funds is the S&P 500. This is a list of 500 companies with the highest market capitalizations ("cap" in stock terms), meaning the biggest volume of money in the stock market. If you read a list of these companies you'll see a lot that have been around for decades. A very different creature is the Russell 2000 index, which is composed of the
smallest 2000 companies on the Russell 3000 index, making it a small-cap index. The Russell 3000 index is pretty close to a total stock market index, since it contains 98% of publicly traded companies. You can also find mid-cap indices that are usually based off of the bottom 800 of the Russell 1000 (top 1000 companies). There are a variety of other indices, but in general if you don't recognize the index they're using, it's probably a really specific selection of stocks, and could be very volatile. Look it up before you buy.
The point of index funds is for the fund manager to just give up trying to do better than the stock market, which is hard and expensive for investors because it increases the amount of buying and selling of stocks in the fund. Actively managed funds usually perform slightly worse than expected based upon the general market performance because of their higher expenses. Sometimes they perform much worse than expected because the manager makes a wrong guess about where the market is going. Index funds say, "We're not even going to try to beat the average," and just invest the fund in a pool of stocks designed to mimic the index.
Sometimes it's hard for the fund manager to exactly mimic the holdings of an index and the index fund might drift, which would lead the fund to perform differently from the index it's based upon. It's actually bad if your index fund beats the index! This means it's not mimicking the index properly and probably has a different composition than you expected when you bought it, and next year it might well underperform the market. Generally when you buy an index fund you want one that has performance in line with the index it is based upon and lower fees than alternative candidates (if two funds are supposed to mimic the S&P 500 they should perform the same, so you should pick the one with the lowest fees, everything else being equal).
When people talk about investing in index funds, they're usually talking about either investing in an index fund based upon the S&P 500 (S&P 500 index) or a total stock market fund (Russell 3000 index). Mid-cap and small-cap indices can have their place, but it's usually not as the foundation of a new portfolio because these smaller companies might have more room to grow than established large cap companies and could take off like blazes, but they can also crash and burn.
SPY S&P 500 is a similar but different beast. It's an exchange-traded fund, which is bought and sold on the market like stocks, and is designed to mimic the S&P 500 like an S&P 500 mutual fund. I won't say much about it because I don't know a lot about ETFs and don't own it, but when I was looking at it before I found it to have low expenses but to be more expensive for me to buy in because of a trading fee (I don't trade in individual stocks so don't have a good account set up at a brokerage for this). It depends on how cheaply you can buy it, I guess.
As for paying off the mortgage vs putting it in your Roth, paying down your mortgage last year would have given you an effective return on your money of 3.75%, while the S&P 500 brought in 29.6%. Of course in 2008 it was down 30.0%. Investing in the stock market is generally more profitable than paying down a very very low interest rate debt, but you never know what's going to happen.
Incidentally that's also why I would recommend putting the money into savings and a Roth. You need to have a lot of flexibility as the sole provider for a family. You can withdraw your contributions to a Roth if you absolutely must, and having money in cash savings is the ultimate in flexibility. If you put it in the house, it's stuck there in a single asset and you have to refinance, take out a home equity loan, or sell if you need to get your hands on that money.
What's your emergency fund target? You should probably stash at least 6 months of expenses, maybe more.