"He sat vigilantly beside his sick father." Here, 'vigilantly' is-

• He sat vigilantly beside his sick father.
→ Here, 'vigilantly' is- an adverb.
→ উক্ত বাক্যে verb হচ্ছে 'sat' আর কীভাবে বসেছিল বুঝালে 'vigilantly = সতর্কভাবে' বোঝায়।
→ এখানে, 'vigilantly' adverb 'sat' verb- কে modify করেছে।
→ অর্থাৎ, যে word কোনো verb- কে modify করে, তাকে Adverb বলে।

• Vigilantly (adverb)
English Meaning: in a way that is always careful to notice things.
Bangla Meaning: সতর্কভাবে।

Example Sentence:
1. I’m watching the situation vigilantly.
2. We must vigilantly patrol our coasts and airspace until further notice.
3. He quickly finished his homework before heading out.
4. The athlete ran swiftly to the finish line.
5. The professor speaks passionately about the subject.

Here, all underlined words are 'adverb'.

Source:
1. Cambridge Dictionary.
2. Applied English Grammar and Composition by P C Das.