PKCS#1 is the first standard published by RSA Security, which was setup by the inventors of RSA, Rivest, Shamir and Adleman. It describes how the RSA problem can be used to perform secure signature generation and encryption.
PKCS#1 is generally used when it comes to describing the padding mechanism of RSA. PKCS#1 however also describes the modular exponentiation and integer to octet string conversion (I2OSP and OS2IP). Furthermore, the PKCS#1 standard lists the possible hashing algorithms that can be used for signature generation. Finally it contains an ASN.1 specification for the various data structures used.
Currently version 2.1 and 2.2 of PKCS#1 are mostly referred to. Version 2.0 added the OAEP encryption algorithm and version 2.1 the PSS signature generation algorithm. It however also cover the PKCS#1 v1.5 padding scheme for backwards compatibility. Version 2.2 mainly completes the support of the various SHA-2 hash algorithms. Wikipedia has a list with the changes within the various versions.
The PKCS#1 v1.5 padding schemes for signature generation and encryption differ. This can introduce confusion, programming errors and vulnerabilities if the schemes are used without knowledge about the schemes or cryptographic API. The PKCS#1 v1.5 padding and signature scheme is deterministic while the PKCS#1 v1.5 padding scheme for encryption is non-deterministic (relies on a random number generator). The PKCS#1 v1.5 scheme for encryption can be vulnerable to padding oracle attacks.
OAEP and PSS have a security proof that states that OAEP and PSS are secure given that the underlying primitives (RSA and the hash algorithms) are secure.
PKCS#1 does not offer support for deterministic encryption or homomorphic encryption. Neither does it indicate how the key generation should be performed or how the modular exponentiation function can be implemented or secured.