RSA is, as you may probably know, the most widely used public key cryptography algorithm. It can be used for signing and encryption, RSA-PSS is about signing (something similar, RSA-OAEP, exists for encryption, but that's not my main topic).
The formula for the RSA-algorithm is S = M^k mod N (S is the signature, M the input, k the private key and N some big prime number). One important thing is that M is not the Message itself, but some encoding of the message. A simple way of doing this encoding is using a hash-function, for example SHA256. This is basically how old standards (like PKCS #1 1.5) worked. While no attacks exist against this scheme, it's believed that this can be improved. One reason is that while the RSA-function accepts an input of size N (which is the same length as the keysize, for example 2048/4096 bit), hash-functions usually produce much smaller inputs (something like 160/256 bit).
An improved scheme for that is the Probabilistic Signature Scheme (PSS), (Bellare/Rogaway 1996/1998). PSS is "provable secure". It does not mean that the outcoming algorithm is "provable secure" (that's impossible with today's math), but that the outcome is as secure as the input algorithm RSA and the used hash function (so-called "random oracle model"). A standard for PSS-encryption is PKCS #1 2.1 (republished as RFC 3447) So PSS in general is a good idea as a security measure, but as there is no real pressure to implement it, it's still not used very much. Just an example, the new DNSSEC ressource records just published last year still use the old PKCS #1 1.5 standard.