When you're insensitive, you're not feeling something. You can be insensitive to the weather or other people's problems.
This word has two meanings that are closely related. When your foot is asleep, it's insensitive or numb — you can't feel your friend poking at it. When you get really cold, you can become insensitive to pain. In the other sense, insensitive means the opposite of caring and sympathetic — you're insensitive to other people's feelings. If you make a joke about your friend's bad haircut when you know she's embarrassed about it, you're being insensitive.