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 cannot store the passphrase for your private key permanently using Pagent, PuTTY's SSH key agent. You will have to store it unencrypted if you wish to avoid retyping the passphrase again and agian.
I suggest protecting all your keys with one standard easy to type passphrase.
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.)