You need to ask for the apartment number too, which is often preceded by a # symbol. Many apartment blocks are called by a number rather than a name, so you have two sets of numbers, one is the house number and the other is the apartment number.I noticed I forgot to mention we also ask the house number.
When handwritten, the apartment number goes at the right with the house number on the left of the same line (e.g. 8020 West Maryland Ave #123). However, it's usual in the US to have an optional second line to the address on web-based forms (in additional to normal address line, city, state and zip) and the apartment number either as Apt NNN or #NNN goes on that line, if applicable. I know it seems strange to a European, but - yes - that means the apartment number really is two lines after the recipient's name on the address label, and not the first line after.