How would you identify the period, block and group of the element with the electron configuration `[Ar]3d^7 4s^2`?

The symbol `[Ar]` in the condensed electron configuration of this element resembles the electron configuration of a ground-state argon `Ar` atom. It represents all inner shell electrons of this atom. (see this problem for details about condensed electron configurations.)

On the top of that, a ground-state atom of this element would contain valence electrons `3d^7 4s^2`. It would lie in the period right underneath the one containing argon. Argon is the last element of the third period. As a result, this element is in the fourth period of the periodic table. (And hence `"n" = 4`)

The block on the periodic table an element belongs to is dependent on the type of the occupied electron orbital of highest potential energy.

Referring to the Aufbau Diagram above, the electron of the highest potential energy in a ground-state atom of this element lies in a `4d` orbital (the one in an orbital that is reached after all other occupied atomic orbitals.) That element, therefore, is located in the `d` block of the periodic table.

An atom of this element contains `9` valence electrons in the ground state. It would thus be in IUPAC Group `9` of the periodic table, which corresponds to old IUPAC Group `"VIII"`.

References:
"Group (periodic table)", English Wikipedia, https://en.wikipedia.org/wiki/Group_(periodic_table)

"Block (periodic table)", English Wikipedia,
https://en.wikipedia.org/wiki/Block_(periodic_table)