Do any one-key-of-many cryptographic schemes exist? I'm pretty sure I understand how public/private key cryptography works. Anybody can encrypt a message using a well-known public key, but only the person who holds the private key can decrypt it.

My question concerns securely sending a message to multiple people. Say there are n public/private key pairs. Is there a way to encrypt a message using the n public keys such that any one of the private key holders can decrypt the message in its entirety, but nobody else can?

Further, unrelated to public/private key cryptography, is there any scheme where there are n shared keys such that anybody who knows the n keys can encrypt a message such that any one of the shared keys can decrypt the message in its entirety, but nobody else can?
Yes, absolutely.  Here is the standard construction to address this problem.  Let $pk_1,\dots,pk_n$ be the public keys of the $n$ recipients.  We pick a random symmetric key $k$, encrypt the message $m$ (using authenticated encryption) under key $k$ to get $c=AE_k(m)$, and then encrypt $k$ under each of the public keys. Finally, we form the whole ciphertext as

$$E_{pk_1}(k),E_{pk_2}(k),\dots,E_{pk_n}(k),c.$$

That's what we treat as our ciphertext.  Now any of the $n$ recipients can decrypt (they can recover $k$ using their private key, and then use that to decrypt $c$), but no one else can.

The same method works in the shared-key (symmetric-key) setting as well.

This standard construction is standardized in more detail in "RFC 4880: OpenPGP Message Format" and in PKCS#7 (enveloped-data).  (Thanks, David Cary and DrLecter!)

You might also enjoy looking into broadcast encryption, which solves a related problem.  It basically allows you to transmit a message to millions of receives, without having a huge blowup in the size of the ciphertext (so it achieves much shorter ciphertexts than the simple scheme above), though there are some details and a little bit of additional coordination required.

For digital signatures, you might enjoy looking at ring signatures and group signatures.