I'm logging into a server that I own with PuTTY on Windows. Is there any way that I can save the passphrase for my private key (like I can in the Keychain on my Mac)?
You can store the passphrase for your private key, for the current session, using Pageant, PuTTY's SSH agent:
But you cannot store the server password itsef:
(See also PuTTY wishlist: remember-password)
On Windows, I have my private key passwordless but encrypted using the EFS, so it can be loaded at startup automatically but only if I log in with my password. (Previously I had it in a TrueCrypt volume, with that volume's keyfile on an USB drive. Don't ask me which is more secure, if at all.)