Can somebody please help me. I don't understand how to even solve this problem or even get the correct answer.?

The IUPAC name of the compound : 1-ethyl-2-methylcyclopentane

The parent chain is a cyclic saturated hydrocarbon with 5 C atom, i.e. a cycolpentane ring.

There are 2 side chains, namely a ethyl group and a methyl group. They are arranged in aliphatical order, i.e. "e" before "m". They are in the positions 1 and 2, and thus 1-ethyl-2-methyl- is preferred.

The IUPAC name is thus 1-ethyl-2-methylcyclopentane. According to the rules, use "-" between a number and an alphabet, use "," between two numbers, and there should be no space in the whole name.

1 ethyl, 2 methyl cyclopentane

Determine the cycloalkane parent, in this case the cyclic part is 5 carbons, hence cyclopentane.
Note that if one of the side chains had more carbons than the cyclic portion of the molecule, it would be named as the longer chain, not a cyclo....
Number the carbons on the ring starting with the longer one and working toward the nearest ones. If you're not sure which direction to make the numbers, number them so that the numbers of the numbered carbons are the lowest possible number when added together.
name from 1 to 2 to 3rd, etc..
so 1 ethyl, 2 methyl cyclopentane.

The ring of five carbons, with single bonds to other carbons (and to two other single bonds, unless otherwise posted) is called a cyclopentane.

cyclo = a ring (a cycle)
penta = five (from the Greek)
pentane is a 5-carbon carbohydrate

then you have
CH3 = a methyl
CH2CH3 is an ethyl (see below)

The name will look something like

(1, 2) cyclopentane methyl-ethyl
or
Cyclopentane 1-ethyl 2-methyl

(means the same thing)

The two consecutive numbers (1 and 2) show that the links to the cycle are consecutive.

cyclopentane is the main part = the ring of 5 carbons
methyl = CH3
ethyl = CH2CH3

Another version is

-----
CH4 is a methane molecule.

If you remove one hydrogen in order to liberate a bond (so that it can be tied to some other molecule - as is the case here) it becomes an ethyl (-CH3) which has a free bond at the "C", represented by the dash.

Similarly
CH3CH3 is an ethane molecule.
Remove one H to liberate a bond, and it becomes a radical
-CH2CH3 = ethyl.

The dash (the open bond) is then used to attach it to the cyclopentane.

The numbers 1 and 2 in the formulation, simply indicate that the places where they link are consecutive.

It could have been 4 and 5 (who cares) except that we always begin with 1 where there is the first bond that is used by some outside molecule (other than a simple H).

In the ring (the pentagon), it is common to NOT show C nor H; you have to know this from basic organic chemistry.

Each angle is a C. C has four bonds. Each single dash uses up two bonds. Whatever remains (and is not indicated by something) is an H.