I have a private laptop and have some Evernote notes on it. Is it safe to sync the notes to a public computer running Windows 7, where I am not the administrator(I am a common user with my own password)?
It depends how much you trust the administrator.
If the administrator wanted to, s/he could easily install a (software-based) keylogger, steal your password and with it access all your files.
And because most keyloggers aren't easily noticed, you may not know your being keylogged.
Anyone however could install a hardware based keylogger.
If you do not trust your local administrator, encrypt all sensitive data. Evernote supports this. As I posted in response to @wizlog's answer, on a windows computer malicious admins don't can Run->Cmd net user USERNAME PASSWORD to reset any other local user's password.
They don't even need to go through the hassle of logging your keystrokes or anything involved. It's literally a one-line command.
However, if your notes are encrypted they will not be able to access the information.
As a side note, if you are an Evernote Business user, know that any information stored in "company" notebooks belong to the organization and can be accessed at any time by account administrators.
The encryption functionality leaves a lot to be desired. You can't encrypt entire notebooks, only single notes. I think they're working on that. Supposedly the encryption is also on the weak side.
You can encrypt things yourself before attaching them to a note, but that will hinder usability in some cases i.e., encrypted PDF content will not be available via search.
Edit: You should probably check out this Ars Technica article if you're planning on saving sensitive data in EN.