I am new to LINUX and am trying to install Fedora 20 on an ACER Aspire Z5600, the specs are as follows : Pentium Dual Core 64-bit 2.7 GHz processor, 3 GB RAM and well over 100 GB of free HDD space.
When it starts I get the message: not enough memory to load image.
Not sure what the problem is.

Everybody is, once.
That seems like it would be more than enough. Hmmm.

So you installed it, it installed cleanly and when you went to boot it (was this the first boot on a new install?) the whole mess crapped out over space? Can you post the complete error message?
Also, just between amigos, have you been messing around with your boot-loader? It's cool, I mean, I know all about the grubby-grub thing but if that's what you were up to when this thing went ass over tea cups, just cop to it now, it will make it easier to fix.

What is the image (and size)?
Or is this a default desktop image?

The problem is probably the BIOS not reporting the correct/full memory. Seems common on ACERs.

Before hitting <Enter> on the CD boot screen, hit the <Tab> key - it wil give you the kernel options for the installer. Go to the end of the line and add the following (note the space character at the beginning), and hit <Enter>. After the install completes all your memory should be recognised correctly (not just the 1024m)As an aside, there is very ittle to be gained installing Fedora 20 at this time - F21 has been released, and you should probably install that. Fedora has a fast release cycle, and you really need to staay current.