一阵子 is an informal noun meaning 'a short period of time' or 'a while'. It often refers to a concrete, countable chunk of time (e.g., 休息了一阵子 'rested for a while'). It can also describe a brief episode of emotion or activity (e.g., 一阵子忙 'a burst of busyness', 一阵子气 'a fit of anger'). Note that 一阵子 is more casual than 一段时间 and is especially common in spoken Chinese.