Originally it used to mean "quantitative", as opposed to "qualitative". It is a type of investment strategies where you target to be right "on average". Once you are right, say, 51% of the times, you aim to reach this asymptotic behavior by trading more and more. Either horizontally (trading more stocks) or vertically (trading more often). You want to keep the law of large numbers on your side to consistently earn that 1% edge that you found.
This contrasts with qualitative investment strategies (sometimes called "discretionary") where you target very precise and punctual events on which you have a very high (say 90%) chance of being correct.
You can imagine both these strategies with a coin toss game.
With a quantitative approach, you would try to find something that allows you to have just even barely more than a 50% chance of winning. Once you find that, you want to bet as much as possible. This is close to the strategy of a casino: they have games which all have an ever so slightly positive expected value, then they just have to make a lot of people play these games.
With a qualitative strategy, you would study very hard to find an event when you can predict at 90% chance the result of the coin toss. You don't play the game until this event is about to happen, and you bet big once it is about to happen.
Nowadays, the term "quant" has a broader meaning, which roughly encompass any kind of financial work which is heavy on math, or sophisticated. You can even find "back office quants", "pricing quants", etc
Algorithmic trading here means more about how to "execute trades" thus reduce the "execution cost" of a trade, rather than using quantitative method to gain advantage (alpha). They are usually employed by firms executing large trades, no mather whether the trade comes from a traditional trading form, or a quantitative trading firm. There are definitely overlaps, especially for HFTs regarding market microstructure, but in HTFs case, the market microstructure is built-in the HFT strategy itself, while for algorithmic trading can be decoupled as execution from the actual trading strategy that produces the majority of the "alpha"s.
I think it’s more or less the same idea, quant trading typically refers to people doing quantitative analysis as part of a larger strategy, and algorithmic trading is all about actually executing trades automatically. Definitely significant overlap
