While I appreciate the answers of the ArcGIS power users here, I do think they overrate what is really necessary. ArcGIS for Desktop, unless you activate 64-bit geoprocessing, can't even adress more than about 3 GB, and I have seen it crash when it attempted to do so in extensions that didn't fully catch this limitation in their code...
I have seen ArcGIS 10.0 run on a 10 year old single core 1.5 GHz with just 1.5 GB RAM. Was it a pleasure to use? No, but it wasn't entirely unusable either.
I recently installed 10.3 for a friend on a dual core Acer laptop of five years old. Back then, except for a mediocre 64bit Turion X2 processor, this laptop was quite well equipped with 3 GB RAM, a 500 GB harddisk including free secondary hard drive bay, and a 1 GB graphics card. I have done other work on this laptop before, including upgrading it with a 128 GB Samsung 840 Pro SSD installed with Windows 7, and putting the hard drive in the secondary bay for data storage. Adding an express card USB 3, this means this laptop is actually quite capable of pumping data around, especially after I completely dis- and re-assembled it to replace the dried out cooling paste of the main processor and graphics card.
Start up time of ArcGIS on most systems nowadays is longer than the startup time of the operating system, and with complex map documents, it will add up.
On the specs of the laptop above, opening a rather complex map document can take up to 2 minutes, once you are in though, the interface is quite useable. Opening ArcMap itself, without a saved map document, takes about 55-60 seconds. Slow, but bearable... To put this also in perspective: on my own 2nd generation Core i5 3.0 GHz quadcore desktop with 16 GB RAM and 500 GB operating system SSD, it also still takes about 25 seconds to fire up ArcMap, so I don't think the 60 seconds is outrageous in this perspective.
So, my verdict:
- Will you be able to run it on this laptop? Probably yes.
- Will you need patience now and then? Absolutely, especially opening a complex map document.
- Will it be a pleasure all the time? No, but bearable if enough patience.
- Will it be able to do everything and complex geoprocessing on large datasets? Not in all cases, but you should be able to do quite a lot on smaller / medium sized datasets and / or less complex geoprocessing operations and with enough patience to sit out processing time.
- Would I recommend this configuration? No, get something better if you can afford it, especially with a tad more RAM, like 4 GB minimum.
Lastly: forget running multiple ArcMap sessions / applications on this configuration. On my own desktop with 16 GB RAM, I have had up to 4 ArcMap sessions open concurrently, with 3 running batch geoprocessing operations for days on end against their own hard drives connected via USB or internal, and the fourth session used for "normal" work preparing a map document. This has proven reliable, but you can forget doing such multi-tasking on the suggested laptop configuration.