I have been told it signifies the "security parameter of the cryptosystem" but this is not clear to me. In this picture, you can see that
$\lvert p \rvert = \mu$ bits
$\lvert q \rvert = 2\lambda - \mu$ bits
which means $\lvert \Delta_K \rvert = 2\lambda$ bits
So does all this mean that if I want 128 bits of security, $\lambda$ has to be 128 and $\Delta_K$ 256 bits?
If yes, this would not make sense with their results they mention and also all the work I have done with this system. For a 128-bit security in this system, the size of $\Delta_K$ has to be 1828 bits. This is something I can strongly assure you of.
Any pointers would be greatly appreciated. Lastly reiterating the question,
"What does the notation $1^\lambda$ mean when it is passed to KeyGen(), Encrypt() and Decrypt() ?"