Obviously, to make some PS1 homebrew you'll need an SDK to be able to develop software for the platform in the first place. The SDKs that are going to be covered in this tutorial series is the official Programmers Tool SDK (often known as PsyQ) and PSn00bSDK.
There are other open source SDK projects but with varying degrees of programming quality and hardware support and usually have drastically different library APIs which are not compatible with this tutorial series. If you much rather use an open source SDK I recommend using PSn00bSDK as its the best one so far (not a shameless self plug, its actually better than others) and follows more or less the same API syntax as the official SDK making it very compatible with this tutorial series.
This is the official SDK provided by Sony and SN Systems used by pretty much all officially licensed developers back in the day, though there were a number that used Codewarrior for PS1 instead. Its often called PsyQ SDK but the correct name is Programmers Tool SDK as that is what it was officially called in the distribution discs and manuals, and the PsyQ name also exists in the PC compatible versions of the Ultra64 SDK.
Majority of the SDK such as tools and libraries was developed by Sony themselves but the toolchain (SDevTC) was made by SN Systems and is based off of the GCC toolchain. But the object and library file format appear to be of a custom format and do not appear to be compatible with the standard GCC file formats, the toolchain additionally comes with custom linker and library tools.
The biggest advantage of using this SDK is that you have all console functionality readily accessible out of the box, but the biggest downside is that while Sony no longer seems to care about protecting the official SDK it is still technically a legal gray area to use it for homebrew. While people have distributed commercial homebrew made with the official SDK and never seem to have gotten in trouble for it, I'd say you can still use this SDK for non commercial projects but that's entirely up to you, not me.
A lot of people say this SDK only works in 32-bit versions of Windows due to its age and that you have to use a VM running XP to be able to use it on a modern PC running 64-bit Windows. While many of the command line tools were only distributed in 16-bit DOS versions there are Win32 versions of the toolchain that would would work in 64-bit Windows so the ability to compile programs would still be possible. The cpe2x tool can be replaced with Orion's version that's built for Win32 and bmp2tim can be replaced with the much more flexible img2tim tool. There are no Win32 versions of the RSD/TMD tools for 3D models unfortunately.
This is an open source PS1 SDK project of my own creation and was used to create a small demo called n00bdemo, which demonstrates the graphical potential that PSn00bSDK is capable of wiht full GTE support. As far as I'm aware, this is the only open source PS1 SDK capable of pushing out graphics as competently as the official SDK (especially the GTE support).
PSn00bSDK's library API follows the same API of the official libraries for familiarity with experienced developers and to make porting existing homebrew originally made with the Programmers Tool/PsyQ SDK to PSn00bSDK easier. The main goal of the PSn00bSDK project is to create a free and open source PS1 SDK solution that has the potential of replacing the official SDK with eventual support of all the hardware features of the console.
The biggest advantage of PSn00bSDK is its 100% legal to use and is by far the most powerful as far as open source PS1 SDKs go. The SDK project is licensed under MPL (Mozilla Public License) which shouldn't be a deal breaker to those looking into making commercial homebrew with PSn00bSDK as the MPL license only applies to PSn00bSDK, not the project using PSn00bSDK. You may want to look up the license yourself if you want to know the exact details of how it works however.
One of the downsides of PSn00bSDK however is its a bit tricky to setup compared to the Programmers Tool SDK especially to those working in Windows. If you're new to programming for the PS1 I recommend sticking with the official SDK for educational purposes unless you're a seasoned Linux user who knows how to compile things.
There are two Programmers Tool SDK archives floating around the Internet. One is a very old dump that leaked many years ago and is the version that is most commonly found. The other one and is one I recommend using is a dump of the Programmers Tool CD 2.2 from 1998 and is much cleaner than the old dump. Most importantly it comes with Win32 versions of the library/linker utilities among other things.
The old PsyQ dump however includes some tools not available in the Programmers Tool CD such as newer versions of ccpsx, cc1psx and cc1plpsx compilers and BUILDCD. Though I don't think BUILDCD is all that useful anymore now that there's MKPSXISO that's newer, faster, easier to use and works on modern operating systems (including Linux and BSDs).
Since the installer program in the Programmers Tool CD is a 16-bit application, installation of the SDK had to be done manually.