There is two ways to answer this question: from your question point of view (where linux can be) and from more general point of view (different types of hardware).
Below I am trying to answer your questions:
Your normal computer. In that definition includes also laptops.
High-end (typically professional) desktop computer, including more memory, better processors and so on. May include added redundancy (for example dual power source) too.
Computer providing services to other computers, for example HTTP service or email service.
When talking in context of this question, server can be any hardware from laptop, desktop to dedicated server hardware, including for example rack mounting and more reliable power source and better components.
Dedicated server hardware is specifically designed to be more reliable and have higher performance. Typically there is for example no powerful graphics card, no sound card and so on.