정신없다 means you're beside yourself from a shock, or extremely busy with too many things to take care of.
You could also use it to describe doing something briskly without paying attention to one's surroundings, as in 정신없이 일하다/먹다/자다/달리다, etc.
Here are some sentence examples.
- 엄마는 동생이 다쳤다는 소식을 듣고 정신없이 달려가셨다.
=> Mother ran off like crazy at the news of my brother's injury.
- 이번 주는 분기말이라서 회사일로 정신이 없어.
=> I'm swamped at work as this is a quarter-end week.
- 학기말 시험으로 정신없다 보니 일주일이 어떻게 지나갔는지 모르겠어요.
=> I don't know how the week went by as I've been so wrapped up in my final exam.