I assume you are referring to a laptop. If so, yes, most laptops can turn on without their battery as long as the power cord is plugged in.
Desktops have a CMOS battery, but that is also not essential to turning on the machine - it stores a few motherboard related settings - but if you don't customize your BIOS settings, you likely won't notice the difference aside from needing to hit F2 to start the computer due to a warning message.
Other “computers” such as phones and tablets, for the most part will not function without a battery, even if plugged in (there are many that will) - but then, even these devices can technically be turned on with a power source connected to the battery terminal internally.
On the very technical side of things, just about anything that uses a battery could be rigged to run off of any other power supply. Manufacturers can and sometimes do implement measures to prevent such capability - but the vast majority of electronics you can get your hands on could be rooted rigged in such a way.
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