The full Ubuntu Desktop image contains everything you need to turn a Raspberry Pi into your main PC, from surfing the web and writing documents to developing software. Because of its size, it only works on the Raspberry Pi 4 models with 4GB or 8GB of RAM.
32 bit vs 64 bit
The Raspberry Pi 2 only supports 32 bits, so that's an easy choice. However the Raspberry Pi 3 and 4 are 64 bit boards. According to the Raspberry Pi foundation, there are limited benefits to using the 64 bit version for the Pi 3 due to the fact that it only supports 1GB of memory; however, with the Pi 4, the 64 bit version should be faster.
Note: Raspberry Pi Zero 2 supports both 32 bit and 64 bit images. Currently only 32 bit images work out of the box, however you can also use 64 bit images with a workaround detailed here.
The Ubuntu Server image is much smaller than the Desktop version, although you can install flavours of the Ubuntu Desktop on top of it. It gives you access to the Ubuntu CLI and by extension, all of the latest open source. Ubuntu Server works on the Raspberry Pi 2, 3 and 4.
Ubuntu Core is a leaner, containerised operating system built on snaps, our universal packaging format. It supports Secure Boot and Full Disk Encryption for added security as well as OTA updates. Develop your IoT software on your Ubuntu Desktop and deploy it to an Ubuntu Core device with ease, even at scale.