Background (in my limited understanding):
The amiga's native RGB monitor output is at 15.5kHz. PC monitor frequencies are generally somewhere between 30 and 100kHz.
Original amiga monitors accepted video input at the frequeny the amiga produced - 15.5KHz. Later 'multiscan' monitors accepted frequencies between around 15.5 and 50 KHz, allowing both PC and amiga to share the same monitor. Modern PC monitors are generally some range between 30 and 100 kHz, so 'original' amiga screenmodes will not be displayed on them.
The amiga is capable of producing screenmodes at 30khz (-ish), but it takes its toll on processing (and, bizarrely the serial port baud rate).
Step in the scandoubler, stage left: the scandoubler takes the amiga's native 15.5kHz signal and doubles the scanrate to 31kHz with no extra overhead on the amiga's chipset.
You may know all this already, so apologies if you feel patronised.
To change screen settings:
Go into your workbench drive, prefs drawer, double-click screenmode. The graphics card you're using in your mediator is a voodoo (3, 4 or 5 - it makes little difference at this point).
You want to choose one if the screenmodes which start with VOODOO, eg VOODOO:1024x768:32-bit - will give you 32 bit (aka truecolour) 1024x768 resolution through the mediator/voodoo card. I've found 24-bit (aka high colour) works quite a bit faster, and the colour difference is not really noticable.
If you choose a screenmode labelled with just the numbers (eg 640x320) this will display through the amiga AGA chipset (it will output the video signal through the 23-pin RGB connector next to the composite out port. If you have a scandoubler attached it should output these 'native' AGA screenmodes through the scandoubler port aswell at twice the frequency, so normal 15.5KHz screenmodes will come out as 31KHz.
There may be other monitor modes available aswell (eg starting with DBLPAL, PAL, NTSC or DBLNTSC): PAL, NTSC you may be familiar with; DBLPAL DBLNTSC are the ~31kHz screenmodes produced by the amiga chipset (as discussed above) - note these will NOT be doubled to 62kHz by the scandoubler - they will just be passed through. they will still exact a heavy toll on your chipset though.
Hope that helps, and is not too inaccurate.