阵子 is a measure word for a period of time, often used in spoken Chinese. It pairs with 一 (yi zhèn zi, 'a while'), 这 (zhè zhèn zi, 'these days'), or 那 (nà zhèn zi, 'those days') to refer to a non-specific but contiguous stretch of time. You can also add 了 to indicate duration (有一阵子了 'it's been a while').