If something happens normally, it happens in the usual or expected way.

Are the phones working normally again? 这部电话又能正常使用了吗？

If you normally do something, you usually or regularly do it.

She doesn't normally arrive until ten. 她通常不到10点不会来。

Normally, I plan one or two days ahead. 通常我会提前一两天作出计划。