Say you have a Access Point (router on the end of a Ethernet cable) in another location to beam out a secondary WiFi signal.
Or you have say a powerline adaptor, which takes the signal via your mains, and beams out a WiFi signal.
Or you have one of these very overpriced at +£100 each unit, mesh systems.
Are they actually doing anything different to each other?
In terms of the Wi-Fi, basically no - and Access Point is an Access Point is an Access Point, whether it is a separate stand alone device, built into a SOHO router or a HomePlug, or a bit of software running on a PC or a phone running something often styled as a "personal hotspot" which turns it into an AP. Or whatever.
Different AP's offer different facilities - there's a lot of variations and options in the Wi-Fi standards - but fundamentally they are all basically doing the same job.
Where the solutions you describe differ is in how they establish what I call the "backhaul" link between AP(s) and the rest of the network. It could be wired ethernet (best way) could be HomePlug (tunneled over the mains) or could be Wi-Fi for backhaul as well as client access which is what a lot of the newer "whole home" and so-called "mesh" systems do.
Would a phone, for example still be the same, and hold onto a mesh unit from another room, despite being near another stronger one.
Yes it could - it depends a bit on how "brutal" the mesh system is about encouraging/forcing the client from one AP to another (it's possible they may not do it at all) which is implementation and/vendor dependent. But fundamentally "mesh" AP's and just AP's like any other, the don't possess some "magic" woo woo the makes them "different" to a fleet of AP's with ethernet (or HomePlug) backhauls. Generally the "mesh" part of so-called mesh systems is about them using Wi-Fi for their backhaul links instead of you having to install cables (or HomePlugs.) And the automation probably sorts out which AP "meshes" with which so you don't have to manually configure it yourself. IIRC some have a management app that draw you a diagram of who is talking to who.
Or it is totally different and, in some way all one signal so the phone does not need to jump between the devices?
There's actually no such thing as "Wi-Fi Signal" as most people conceive it. (Big Wi-Fi Myth Number 1.) Wi-Fi is not facilitated by some ethereal energy field like The Force or Ley Lines or something. The AP's are thusly not "field generators" like that dish thing protecting the death star in the Return of the Jedi.
Every single Wi-Fi device on the planet - all the phones, tablet, laptops, printers, eftpos terminals, webcams, car alarms, and AP's - everything "Wi-Fi" both transmits and receives radio signals. There's no asymmetry in this either - the radio emitted by your iPhone is just the same as the radio emitted by an AP and is entitled to be just as "loud" (though phones do tend to try and be a bit miserly to eek out battery life.) Wi-Fi is two-way radio like walkie-talkies (or a conversation) not one-way radio like television (or a lecture.)
In each Wi-Fi cell there's basically a rule that "only one thing at a time can transmit." the AP's are but one of those "things." There are rules in the Wi-Fi protocols which describe how to determine whose turn it is to transmit, which try to ensure some level of fairness.
So the AP's aren't spewing out some kind of "signal" (energy field) that the clients somehow piggy back onto. It's just like sound. Just as you do not require some "thing" to generate "audio signal" in order for you to speak, you just speak and you voice carries as far as the local environment permits - same for Wi-Fi, except it uses radio waves instead of sound waves.
What makes an AP and AP is it is the "point" at which you "access" the rest of the (wired) network from your Wi-Fi device. Us IT nerds are not terribly inventive when it comes to dreaming up names for things.!
AP's give the illusion of "Wi-Fi Signal" because they transmit something called a "beacon" usually 10 time per second which advertises their presence (and name.) It's these beacons that things like Wi-Fi scanner apps listen out for and display. As part of the meta data the scanners NIC gets when grabbing the beacons out of the air, Wi-Fi NIC's record the received signal strength (RSSI) and it's this that your scanner app displays, graphs or shows on the "bar" meter. It looks like it's measuring "field strength" as if taking a temperature, but it's not, it's an illusion and because the beacons happen so frequently, it looks like a continuous reading even though it isn't in the same way that TV/Film fools us into seeing moving images made out of frequently updated static images.
Roaming from AP to AP is about maintaining a usable data service, not constantly "hunting for the best signal."
I think I need to have a lie down now. And
@Tempest - I like your avatar, though of course Defender is the best video game ever and don't let any yoofs claim it's FIFA Soccer '98, Halo or some other nonsense.