Synthesizers can be split into different categories based on certain criteria. First of all, there are modular synths and their counterparts, the all-in-one synths. Because there are different parts that make a synth, you can always split that into different modules. Usually mounted in a rack, modular synths can be moved around and shaped however you want. All-in-one models are, as you expect, built in a single block covered by a single case.
Another way to differentiate synths is based on how they essentially work, and this boils down to analog and digital devices. The digital ones are newer and offer more variety, but the analog ones are the old-style and they sound generally more natural.
You can also make the distinction between monophonic and polyphonic synths. Monophonic means that the device can only create a single not at once, but nowadays most synths are polyphonic because that’s the minimum requirement.