HFT = beat other programs on the market. You do low lvl system stuff tweaked to nanosecond. Quantitative trading = use statistics to build models of strategies to trade on (quant research) Hedge fund = a fund managed to hedge bets to minimize losses. And now an institution can have their hands in all of the above.
Actually you can buy pretty much anything and everything in the terminal and get in touch with just about anyone from institutions to regulatory across the world. It's just that you still have to be humanly present to do it. And be a paying customer. Biggest trades happen among friends on IB.
That is the traditional definition of a hedge fund, but lots of hedge funds do nonhedging at all. Simply a designation given to firms who are governed by how they can solicit business from investors. In a way that designates high net worth and can afford the loss. Also limited in how they can solicit business.
HFT is a way of trading where you hold positions for very little time and you make money due to regulatory environment that mandates it.
Quantitative trading is just using math to automate the trading process in some way.
Two Sigma and Citadel both have a hedge fund as well as a securities arm (called ____ Securities), which does both prop trading and client execution (get order from clients which are other funds and execute for them).
Jane Street and some others like Volant also have both prop trading and client execution.
Banks have securities divisions which primarily do market making, prime brokerage (think Fidelity or Etrade for a big institutional investor) and used to do a lot of prop trading but that's been tightly regulated since Dodd-Frank. Most of them also have hedge funds and PWM services that are completely separate.
Jump, Tower, HRT are HFT shops that started with prop trading (using HFT strategies) but now have all opened internal "hedge funds" which use stat arb strategies like AQR and Two Sigma use. They might advertise this as "medium frequency". Probably only partner money in it.
The common thread is all of them try to buy and sell stuff while turning a profit, the differences are how fast (the slower, the longer you hold and the more risk) and with whose money. PWM = 1 unaffiliated rich persons money which gets outsourced to other funds, hedge fund = many rich people or pension funds/whatever plus employee money, prop trading = partner's money, bank = bank's money, long only asset manager = same as hedge fund, mutual fund = many poor people's money.
A hedge fund is a like a mutual fund but with very little restrictions on what they can not do and as such, are only open to entities that bring in a significant account of capital with a high risk, high reward profile