Cryptographic schemes claim to use a security parameter K for deriving signing key and verification key. What does this parameter signify ? how can be chosen?
The security parameter (usually denoted $k,\lambda$ or $\kappa$) is the input to the algorithms of a cryptosystem against which we measure security. That is it should have no effect on correctness of the scheme but only its security against attacks.
Formally the only requirement for this parameter is that attacks grow in time super-polynomial in this parameter and the time for legitimate applications of the scheme grows polynomially in this parameter.
Practically one also often encounters security parameters to make comparisons between parameter sets across schemes. For example a 384-bit elliptic curve has 192-bit security, AES-192 has 192-bit security and 3072-bit RSA has 128 bit security. So if you are asked for a security parameter independent of the scheme it usually is the "x-bit security" parameter, but when asked for a concrete scheme it usually is the "x-bit $Scheme" parameter. For all practical purposes a general security level / parameter of 128-bit is sufficient.