Just looking to buy a new-ish computer for study for my PhD. I will be using ArcGIS. I know the ideally it would be best to buy all the memory and ram I can afford but I am self-funding on a low budget. I am wondering what's the minimum spec Apple Mac I could use to run this software practically?
If you want to run ArcGIS on a Mac (as I do at home), there are two routes you can take.
The first is to buy virtualization software -- Parallels or VMWare Fusion are probably the best options for that. These allow you to run Windows within OS X, and for each "virtual machine" you can allocate a specific amount of your system's total resources (RAM, CPU cores, and hard disk space). Since it is terribly unwise to allocate all of your system's resources to a virtual machine, I'd recommend a computer with more RAM and CPU cores than you would get if you were just buying a PC. This is the route I chose.
The second route is to use Apple's Boot Camp, whereby you will essentially have to choose whether you want to run OS X or Windows -- one at a time. The upside is that the Windows installation is free to use all of your RAM and CPU, and is also free (unlike Parallels and VMWare Fusion).
Either way, you will need to purchase a standalone copy of Windows, which you will hopefully be able to get through your school for a significant discount -- they're not cheap, at full price!
Having said all this, RAM is the single most important consideration when running ArcGIS specifically, and virtual machines in general. Thankfully, RAM is cheap! (That is, as long as you buy it from crucial.com, newegg.com, or really anywhere other than Apple. It's really easy to install, and Crucial, at least, has a really great tool for helping you find the exact kind you need -- all you do is pick how much you want.)
My recommendation is that you get 8 GB of RAM at a bare minimum, but I would urge you to go up to 16 GB if you can afford it.
I'm running ArcGIS in a VirtualBox virtual machine on a Macbook Pro 2.4 Core 2 Duo with 8GB of RAM and an SSD. VirtualBox is free and works really well and FAST; the VM runs Windows 7 tweaked a bit (lots of online guides about this) to make it bare-bones. I don't run antivirus on the VM either - I just keep a backup of the original Win7 install (+ ArcGIS +tweaks) and restore that if anything goes wrong.
RAM is definitely the priority.
For MINIMAL setup though... I've also got an old Macbook 1.8GHz Core Duo (five year old model now?) with 4GB RAM and a regular platter drive, also with VirtualBox and the same virtual machine. It's a bit slow for any serious geoprocessing operations but ArcGIS works fine for most of the fieldwork I do, and I can usually run a couple of other apps in parallel with the VM.
All that said - I hardly use Arc anymore! Go QGIS ...