What type of chemical bond connects atoms that equally share electrons such as the bonds between a hydrogen atom and a carbon atom in methane?

Covalent bond

Explanation:

If we look at the definition of covalent bond, it states that a chemical bond formed between two atoms when they share equal number of electrons with each other in an attempt to get a stable electronic configuration.

For example: The bond between two hydrogen atoms in a hydrogen molecule where both hydrogen atoms share one electron to get stability (two electrons in valence shell).

Similarly, the example mentioned in the question is of methane molecule with formula CH4, In which each hydrogen shares one electron with Carbon atom that ultimately shares one electron with each hydrogen to make itself and hydrogen atoms stable (you can see attached image showing covalent bond formation in methane) for better understanding.

Hope it helps!