Closet, wardrobe, cabinet, cupboard. Szekrény is a general piece of furniture, which opens to the front and is made for storing objects. If it's in a wall (like "closet" sounds to me), you can differentiate that by saying faliszekrény - wall-cabinet.

This was my path into Hungarian!

To the etymology!
While they look similar (and cover the same meanings), these words seem to have different origins. According to the Wiktionary, szekrény comes from the old French escrin which has its roots in Latin scrinium, meaning "box" or "case".

Meanwhile the German "Schrank" is a word that's only used in the German language space. The Old High German scranc means "weaving, interlacing" and refers to the diagonal lacing structure that was used to hold chests and boxes upright, so they can form a wardrobe. It comes from the Indogermanic root *[s]ker-, "to turn", and the next English relative of that is "curve". (source: Duden Herkunftswörterbuch)

But the similarity is actually nice for learning. :D

Mi van a szekrényben? :D
If you're particularly asking about the wardrobe where you store your clothes (which, I assume, is the actual purpose of a "wardrobe"), you can express this with using ruhásszekrény.

A "cupboard" általában egy kisebb szekrény a konyhában, amelyben csésze, tanyér és más konyhadolog van. A "cabinet" gyakran egy szekrény a különös dologoknak, például boroknak vagy evőedényeknek. A "cabinet"-nek gyakran üvegajtó van.

A "wardrobe" (egy bútor) és a "closet" (a falban) a ruhának.

Minden egyéb szekrényalakat vagy "cupboard" vagy "cabinet" tudnak hívni.

(Remélem, hogy érted a magyaromat.)