What are the metal parts of a boot where the laces are meant to go called? There are the circle parts and the hook parts. Do these have a different name each, or do they have the same name?
The Weinbrenner Shoe Company calls them hook eyelets and "set eyelets:
For shoes and other clothing and accessories, eyelet is the common term:
noun [countable] a small hole in cloth surrounded by a metal ring, for putting a string or rope through [Macmillan]
The more generic term is grommet, though in my experience this usually refers to a larger hole as might be used for a sail or tarpaulin.
It is indeed an eyelet as others have answered. And the part of the lace that you push through the eyelet is called an aglet (the rigid plastic cylinder that binds the end of the lace).
Definition from the Collins English Dictionary:
An eyelet is a small hole with a metal or leather ring around it in cloth, for example, on a sail. You can put cord, rope, or string through it.