-겠 is a tricky one because it can be used in a number of ways with different meanings. As a result, it can frequently trip up Korean learners.
1. -시겠어요? / -시겠습니까? = “Would you ...?”
+ very formal Korean.
(Normally you would use -(으)ㄹ래(요)?
+ the honorific suffix -시- is always used with -겠- in this usage.
2. 겠- = I’m going to ... (expressing intention)
+ formal Korean (or when strongly emphasising something)
(Normally you would use (으)ᄅ게(요))
3. -겠- = I think, I guess, I assume
+This is the most common usage of -겠-
+Gives a nuance that you are somewhat careful with your opinion.
4. -겠- is also used in fixed expressions
= I got it. I understand.
= I don’t get it. I don’t know. I am not sure.