드리다 vs 주다
드리다 is a transitive verb which honors the object of it, not the subject who does the action, so it is different from the ordinary honorific verbs with 시 in them, like 주시다, which honor the person (subject) who does it.
Some other common words in this special group are 여쭈다, 여쭙다(tell, ask), 올리다(present, offer), and 뵈다(see, meet).
These words are therefore only used when the object (recipient) of the verb is older.
For example, if you give a gift to your mother-in-law, you'd say 장모님께 선물을 드리다.
But if your mother-in-law gave you something, you say 장모님이 (내게) 선물을 주셨다.
So 드리다 and 주시다 are used in directly opposite circumstances.
In addition, there are a couple of universal rules in using honorific language:
1. You never apply honorifics to yourself.
-- a) never use the -시- forms referring to yourself.
-- b) never use the recipient-honoring verbs like 드리다 if you're the object of the verb.
2. When you mention an action of the listener toward a third person, use the verb form appropriate to the listener's standpoint.
For example, #1 means a grandfather would never use 드리다 to ask his grandson to bring something to him.
He would just say 줘 (not 주세요 since he's a kid), as in 그거 할아버지(himself) 줘 or 그거 나 줘, not 그거 나 드려.
If he's telling his grandson to bring something to the kid's mother (let's say the grandfather's daughter-in-law), he would say 드리다 speaking from the standpoint of the grandson, as in 그거 엄마 갖다드려라, even though he would never say 드리다 himself when talking to his daughter-in-law.
If the grandfather is asking his son to give something to the son's wife (again the grandfather's daughter-in-law), he would say 그거 네 처 줘라, since a husband and wife are at an equal level.
If we take things even farther, we can even use 드리시다 - I honor someone's action of honoring someone else.
For example, "I" can say 어머니는 할아버지께 인사를 드리셨다 (Mother greeted grandfather) - I say 시었다 to honor mother, who's action is giving something to someone senior to her, so not 주셨다 but 드리셨다.
But the whole honorific system is not rigorously kept up as before owing to the modern small family, so even Korean people are less and less keen on such protocols.
Anyway, you need to know the type of the honorific verb in this case, and the two underlying principles to be able to use the correct honorific language.