In Japanese, あとで is a combination of the noun 後（あと） (behind), and the particle で (with). When these words are linked, they form an adverbial phrase which is used to express ‘after’, or ‘later’. As with で’s standard meaning of ‘with’, あとで is simply highlighting that something will happen, ‘with’ the ‘after/behind’ part of something.
To use あとで, attach it to the end of any past-tense verb, noun followed by の, or verb followed by のは (nominalization).

• 勉強（べんきょう）するのは食（た）べた後（あと）でいいよ。
  You can study after you finish eating. (With the time after eating)
• 仕事（しごと）の 後（あと）で、飲（の）み会（かい）に行（い）きます。
  I will go to the party after work. (With the time after work)
• コピーを取（と）るのは後（あと）でいいから、今（いま）は上司（じょうし）に電話（でんわ）して。
  Since it’s fine to make the copies later, please call the boss right now. (With the time after calling the boss)

後（あと）で may also be used to alter the meaning of an entire phrase, by appearing at the front. This nuance is just that (A) will happen (at some point) after now.

• 後（あと）で洗濯（せんたく）もの干（ほ）してね。
  Please hang the laundry later.