Learn Hardcore Russian: Проверь выключатель, пожалуйста: свет не включается. - Please check the light switch: the light won’t turn on.

Проверь is the perfective imperative (from проверить), so it suggests a single, complete action: check it (once).
Проверяй is the imperfective imperative (from проверять), which more often implies an ongoing/repeated action or a process: be checking / check regularly / check (as a process).
In a practical situation like a broken light, Проверь выключатель sounds natural: do one check now.

Why is выключатель in the form выключатель (no ending change)?

Because выключатель is masculine inanimate, and here it’s the direct object of проверь (imperative of check), so it’s in the accusative case.
For masculine inanimate nouns, accusative = nominative, so it stays выключатель.

What’s the difference between выключатель and свет here?

выключатель = the switch (the physical wall switch/button).
свет here means the light / the lights / the lighting (electric light), i.e., “the light doesn’t come on.” In Russian, свет is often used where English might say the light or the lights.