Exactly. Presumably your SMTP server in your mail program is set to smtp.myhomeinternetprovider.com. When you try and access that server from the library (with an IP address which is provided by the company your library uses for its Internet access), smtp.myhomeinternetprovider.com does not recognise you and refuses access. It's to stop spamming - you could be trying to send out 1000s of e-mails. You won't do this when you are at home because they know who you are.
Receiving e-mail is OK, browsing the web, iChat etc will be OK.
Options:
1. See what SMTP is for the library's Internet provider. You might be able to use it. Probably not.
2. Use a webmail service.
3. Set up your Mac to be its own SMTP server. This is easy and offers lots of flexibililty. Some Internet providers will not allow mail to be received if sent in this manner as your mail will fail in terms of reverse DNS lookup. In my experience, this is only the case for AOL but others might do the same. This was raised
here.
4. Use a .Mac account. You can use smtp.mac.com from anywhere.