An inspector is someone who looks into, observes, and examines things — it's also a word for a detective.
The government employs many inspectors — an inspector has to check out the kitchen of a restaurant to see if they're following the rules for cleanliness. Hospitals need to let inspectors check things out, to make sure conditions are safe. Many types of business have inspectors — especially if there's a health concern. An inspector observes things carefully — an inspector checks things out.