Might

Asking Permission:

May/might we leave now? (Might is less common here and more formal)

Expressing Possibility / Probability:

It may/might rain tomorrow. (There is no difference here between may and might)

Requests:

May we see the menu, please?

In the Past

Might is not the past of may. However, when referring to a statement about probability made in the past it is more common to use might.

He told me that he might be late to the meeting.

When referring to asking permission or making a request in the past it's more common to use could.

I asked him if I could take the day off.

We may/might go as well.

This means that there's a possibility that we will also go; we will go too.

We may/might as well go.

This means that our best option is to go. If nobody has a better plan then we can/should go.