Head Up means that a blip appears on the screen as it will if you look outside, using the stem as a reference point. Useful if tracking and identifying boats moving relative to yourself.
North Up makes the plot look more like the chart, and develops your "eidetic" spatial memory - the invisible map that you carry round in your head.
It's a useful discipline to learn to interpret both modes.
If it's set to true, doesnt it wobble all over the place with your heading? i.e. the image moves around the heading, rather than the heading moving on the image?
True motion is very useful if you want to try and understand a general picture of what's happening. Because true motion actually shows what the echoes are doing as if you were looking at a graphic of a boat moving on a chart it can be easier to spot that something is heading for a particular point (e.g. boats heading towards start of a buoyed channel) so it can give you a better a idea of what's going on. If you're watching some boats heading for the passage round a headland you don't have to adjust for the fact that the headland keeps moving across to the left of your display and down it twice as fast for example - so it becomes obvious all those boats are going to exactly the same place.
The main thing you lose with true motion is the ability to spot a target on a collision course because the target will probably no longer have a constant bearing from the centre of the screen. Having said that a lot of people find it easy to see a constant bearing between two targets on a PPI (Plan Position Indicator - the name often used for displays capable of true motion), especially if you can tag targets on the display
RADAR obviously shows motion (changes in range and bearing) relative to the scanner so it's relative by default. To get true motion you have to subtract your heading and speed which means true motion is only as accurate as your log and compass or your GPS.
Which brings up the next point to watch which is that, if your input is from a GPS, true motion is shown relative to the ground. If your input is from a Walker or paddle-wheel log and magnetic compass then true motion is shown relative to your speed through the water and compass heading, in fact it's actually the speed the log's showing which could be well out and of course compass heading means you need to think of variation and deviation.
As pointed out, if you haven't got a well-damped heading input such as rate-gyro correction on a flux-gate your heading will be bouncing around in true motion a lot too.
Having said that, as an ex-military man, I like true-motion because it does make it easier to see what everyone else is up to - which the military do like to do of course.
Relative motion and relative vectors is used to assess risk of collision.
A relative vector shows the moving relationship between the target and you. If it points towards the centre of the screen, a risk of collision exists.
A true vector gives an indication of the direction the target is going and hence its aspect, but it does not give an indication of risk of collision. Accidents have occurred when one has been mistaken for the other.
I was certainly taught to use radar North up though I have not made that much use of ground satbilised radar, which I would suggest most of the time is not a good idea as you are always reducing the ammount of display available in the direction you are going, Equally except in special circumstances I would avoid offset centre for the similar reasons though it does have more uses in my opinion than true motion which I would suggest only has utility on very large screens.