Not sure what you mean.

TB just downloads those mails that are on the server.

You might specify what you mean by "old e-mails" (and how TB should know which are old), if you have POP or IMAP (with IMAP you could move messages in your Webmail to a separate folder and not subscribing to it in TB).

I use pop.

I have houndreds of emails in my inbox, that ive already red. Ive only used browser email so far, and today Ive installed TB client. When I click in "Get Mail" it starts downloading all my inbox.. And I just want it to download emails from now on, not all the database from the server..

Thanks for you fast reply!

OK, why are you switching to an email client having previously used webmail? What were you seeking to change or improve on? It seems to me that you consider the server to be the right place for your old email.

You say you are using POP. Were you given a choice between POP and IMAP, and if so, why did you choose POP?

It sounds to me as if Thunderbird is simply doing what an email client does when using POP. It asks the server for the available messages. Since it has no sight of your email usage history, it has to assume that all the messages are unread, so fetches them. I suspect you will have an unpleasant surprise when you discover that it has moved all the messages from server to your computer.

So, again, I ask what benefits or advantages did you expect from switching to an email client?

Just like Zenos says, it is difficult for any mail client to know which messages should be downloaded or not.

You may have a look at

http://kb.mozillazine.org/Automatically_download_messages

In

  Account settings > Server Settings

you may experiment a bit: e. g. not selecting "Automatically download new messages".

Be aware that if you *do not* select "Leave messages on server" the messages that TB (any mail client) fetches won't remain on the server (unlike with IMAP).