開(ひら)く is to open something that is shut or folded. 閉(と)じる is the opposite.
ex:
本(ほん)を開く/閉じる
open/close the book
門(もん)を開く/閉じる
open/close the gate
傘(かさ)を開く/閉じる
open/close the umbrella
開(あ)ける is to open something that is closed off. 閉(し)める is the opposite.
ex:
ドアを開ける/閉める
open/close the door
ふたを開ける/閉める
open/close the lid
店(みせ)を開ける/閉める
open/close the shop