Help, Mail(?) not downloading mail

[Daruma Chan]

I'm using Mail to download my e-mails, and these last couple of days, 
I'm having problems with it.

It doesn't seem to download any mails. Or, to be more precise, it 
doesn't seem to register the mail as having been downloaded.

I download my e-mail. Then the next time I download my e-mails, Mail 
would download all the missives I've already downloaded before.

Now my e-mail box is bulging, since the messages are piling up.

I'm using one of those free POP webmail services. Considering that I'm 
having the same problem with two different webmail services, I'm 
inclined to putting the blame on Mail instead of the webmails.

So, is there something I should do?

I'm still on OS 10.1.5, using Mail 1.1

On a related question, where does Mail store the e-mail?

Thanks in advance to all, and please, do have a Merry Christmas and 
Happy New Year