Latency is the amount of time it takes for data packages to go from one point to another on a network, usually in milliseconds. A latency of 400ms, for example, means it takes 0.4 seconds for the information that you used some spell reaches the server, which then has to be processed and the result sent back to you, to then be processed by your computer to display the result. That can be affected by many things, though, like the distance, the amount of hops across the network it takes to reach the destination, and how much concurrent traffic there is both locally and at the destination.
Services used to reduce latency can help with that by using specific routes to optimize how you reach the server (less hops, avoid bad routes, etc.), but how much they affect it can vary a lot. Free ones have way more than likely already been banned by IP here due to them being abused to spam advertisement and such, so you really shouldn't risk it, you will probably just have to wait a few days after sending a ban appeal to get your account back.
In most cases the issue is neither quite on your end (all your connections would be slow in that case) nor on the server (all players would be affected in that case), but on the route your ISP is taking to reach the server. No one has a direct connection to the server, they follow a path over multiple "hops" to intermediate servers that forward the data to the next one, until it arrives. And that's where most issues happen. If any of those hops (which can routinely be 10-15+) is having an issue that slows down their traffic, you will be affected. That's why even someone who doesn't live far from you could have a better connection, it just would take his ISP to be using a different route that is working properly. This is what usually happens suddenly and fixes itself on its own, be it when the issue gets fixed or a new route avoiding that hop is spread across routing tables, but it can take a while.
Your connection speed is also completely irrelevant (well, I suppose if you were on a 14.4k dial-up it would matter, but yeah). You could be driving a Bugatti, but all its speed would be meaningless if you took a street with men at work.