I read somewhere that an SSD can be attached to a motherboard using the PCI-Express bus without a controller. I am not sure what this means. If these are special kind of motherboards, what are some of these motherboards (brand name and model)?
You are probably thinking about SSDs that use PCI Express instead of SATA interface. They can go on any motherboard (provided that there are PCI express slots) and, once system is booted, they behave just as standard disks.
Advantage to this approach is lowering latency and speed increase. Disadvantage is being unable to boot from such device.
It happens that more modern variants of these devices are bootable (they have integrated HDD controller).