Print32-bit software or hardware can only directly represent and operate with numbers up to 2^32 and, hence, can only address up to a maximum of 4GB of memory (that is 2^32 = 4294967296 bytes). If the file system of your operating system is limited to 32-bit integers as well, this also means you cannot have any single file larger than 4GB either in memory or on disk (you can still page or chain larger files together though).
64-bit architectures don’t have this limit. Instead you can access up to 16 terabytes of memory and this is actually only the limit of current chip architectures which "only" use 44 bits which will change over time as software and hardware architectures evolve. Technically with a 64-bit architecture you could access 16 Exabytes of memory (2^64) and while not wanting to paraphrase Bill Gates, that is probably more than we’ll need for the foreseeable future.
There most likely won't be any innate performance benefits to be gained by moving from 32-bit and 64-bit unless you need that extra memory. While in principle, you can move larger amounts of data per time between memory and CPU with 64-bit, this typically doesn't result in significantly improved execution times because of caching and other optimization techniques used by modern CPUs. However if we start using programming models where we run many tasks at once, you might want more than 4GB allocated to those processes. For example if you had 8 tasks that all needed 500MB of RAM each – that’s very close to the 4GB limit in total (500MB * 8 = 4000MB). If you had a machine with more processors (e.g. 64) you would very easily hit the 32-bit 4GB limit as you would only be able to allocate 62.5MB of RAM per processor from your code.
Even with hardware architectures and operating systems mainly being 64-bit these days, a lot of software still is only available as 32-bit versions. 64-bit operating systems are designed to be backwards compatible with 32-bit applications, and if there is no real expected benefit for a particular software, the developer of the software may just as well decide to stick with 32-bit and avoid the efforts and costs that it would take to make the change to 64-bit or even support multiple versions of the software. ArcGIS Desktop is an example of a software that is only available as 32-bit and this is (most likely) not going to change anymore since ArcGIS Pro, which is 64-bit, fills that role. However, Esri also provides a 64-bit geoprocessing extension for ArcGIS Desktop which will be further described in the next section. However, this section is considered optional. You may read through it and learn how it can be set up and about what performance gain can be achieved using it or you may skip most of it and just have a look at the table with the computation time comparison at the very end of the section. But we strongly recommend that you do not try to install 64-bit geoprocessing and perform the steps yourself before you have worked through the rest of lesson and the homework assignment.