A pain is a feeling you experience when you hurt a part of your body or when you are ill. The noun pain can be countable or uncountable. As a countable noun, it can be used with the article a. It can also have the plural form pains.
He complained that he was having terrible pains in his chest.
Sometimes I feel a sharp pain below my ribs.
I have a pain in my head.
I can no longer endure this pain.
As an uncountable noun pain is mainly used to talk about the feeling of being sad or upset.
She couldn't cope with the pain of being separated from her family.
The girl's unfortunate death caused her parents great pain.
Suddenly I felt a severe pain in my head.
This medicine will ease pain.
An ache is a continuous unpleasant pain that is not very strong.
We do not usually use the word ache to talk about a momentary feeling of pain.
Ache can also be used as an intransitive verb. When a part of your body aches, you feel a continuous but not so strong pain there.
Just when I thought I was starting to feel well, my head began to ache.
We usually use the word ache to talk about common pains that we experience in our head, stomach, tooth or back. For example we say headache, backache toothache and stomachache. Native English speakers do not usually say head pain or stomach pain. Note that we can use articles with toothache, backache etc.
I am getting a toothache.