Woody's pretty much right. Typically, for a desktop PC to support two monitors, it will need to have a separate/discrete graphics card (so - not 'on board' graphics) with two graphics ports. These will typically be one VGA and one DVI, or two DVI ports. A very few desktop PC motherboards feature two graphics ports, but they're in the minority.
I'm not sure where Woody's coming from on the memory thing, though. I have regularly and successfully driven two monitors from a low end graphics card (64MB/128MB onboard memory) in a 256MB/512MB machine. However, if you want to do more than office applications/web surfing - gaming, for example, or photo/video editing - then more memory on the graphics card and in the PC itself is definitely a good idea.
No, they absolutely don't. I run a 1,280 x 1,024 monitor and a 1,440 x 900 monitor of very different sizes. With your graphics card, you should get software/drivers that will allow you to set up the monitors how you want and set their resolutions etc. individually. (That's certainly true with NVIDIA cards - I imagine it's equally true for ATI/AMD cards).
Why on earth would your PC shop say that, though? Unless they only have one size of monitor in stock...? :-)