When I play chords, the "D" String buzzes really loudly, but when I play it alone, it doesn't. (Doesn't matter if I pressed the fret or not).
@Kevin Johnsrude is correct, but I will add to his list here:
It is possible for a guitar to have problems with buzzing that are not caused by the guitarist. The action may be too low. The frets may be too warn. The neck may be warped.
Also placing your finger on the string is an art. The target area on your fingers that produces good tone is smaller than that which merely touches the string to the fret.
Sometimes strings can buzz against your finger nails of other fingers on other strings, but you will feel/ notice this most of the time.
Long fingernails on your left hand can also get in the way of proper finger placement (relevant to 5 and 1)
Practice can help improve the accuracy of your finger.
A good guitar teacher would better diagnose your issues, which are likely to be more than one of those listed.
Arch, btw, is the curve of your finger. The more curved your finger is, the more muscle strength you are using (to a degree) and also the more your fingers can select which string(s) they are touching.
You can also add strength to your fingering by pulling back with your entire arm. Avoid gripping too tightly with your thumb, as that can slow you down.