Question: Originally, this was more of a conceptual question that I tried to Google, but after looking through many sources, I keep seeing different definitions and terminologies and a combination of them that I lost track of their actual meanings, and which are referring to the same things:

1. mutual fund
2. index fund
3. mutual ETF
4. index ETF
5. index mutual fund
6. index mutual ETF
7. ETF

Someone told me that DIA is an index ETF that indexes DJI, which makes sense, but all the other terminologies are really tripping me right now. Another source told me that ETFs and mutual funds can be index funds, but they don’t have to be, what's an example of that? And then other sources say that an ETF is a specific type of mutual fund. In any case, I just don't see the big picture with all these technical semantics.

Answer: 
You can simply transfer the money as usual (wire transfer, or any other means). There are no tax implications for moving money into and out of the US, and nobody will care (unless you transport it in cash).

The tax implications are from having that home in the first place. You are required to report on your tax filings all income and property outside the US, although it is of course possible that the house produced no income for you.  

If the house was owned by your mom, and she is gifting you the amount now, this is a *taxable gift*, which *she* - not you - has a US tax liability for, if she is a US person (and independantly, potentially a tax liability in her home country, which you didn't name). 

For a mother-to-son gift, the US tax exclusion is 15000 per year; only money above this is taxable. Note that this limit is per person, giver and receiver, so if either your mom or you are married, both parents could gift each to both of you each 15000 tax free, resulting in 60000 tax free. 

There are no tax consequences of using any amount you happen to have to pay off part of your mortgage.  

Make sure you do it right, as some mortgage companies have funny rules about larger payments - they could consider it a 'pre-payment of your next N monthly payments' and as such just 'safekeep it interest free' for you, and pay your monthly rate every month from it. This is obviously bad for you, and not what you want; you want it applied to the principal immediately, so make sure this is what happens.

Overall, no paperwork, and no tax consequences for you. No worries.
Depending on what country the money is coming from, Transferwise might be an option. If they offer transfers out of the county in question it would probably be the cheapest option in terms of transfer fees. If you use a regular bank you are going to get screwed on fees and exchange rate to the tune of 3 to 5% of the amount transferred. 
Just have your mother make several transfers under 10k each, no tax consequences.
Also, why on earth would you want pay your mortgage early right now? You can probably negotiate something with your bank, so you don't have to pay anything for a couple of month without any impact on your credit. At the very least, look into refinancing at a lower rate, which you probably can, after the recent rate cuts.