An efficiently computable function $f:\{0,1\}^*\rightarrow\{0,1\}^*$ is said to be $(s,\epsilon)$-one-way if for every adversary $\mathsf{A}$ of size* at most $s$ the probability $\Pr\left[{\mathsf{A}(f(x))\in f^{-1}(x)}\right]$ is at most $\epsilon=\epsilon(|x|)$, where the probability is over uniform distribution on the domain and the random coins of $\mathsf{A}$.
The standard assumption one-wayness assumption is that $s=poly(|x|)$ and $\epsilon=negl(|x|)$. A one-way function is subexponentially-hard if $s=2^{{|x|}^{o(1)}}$ and $\epsilon=2^{-{|x|}^{o(1)}}$. Note that the latter implies the former and is, therefore, a stronger assumption.
Similar definitions apply for the other primitives too.
-Geoffrey
*Here, size refers either to the run-time if $\mathsf{A}$ is a probabilistic Turing machine, or the circuit-size in case $\mathsf{A}$ is a circuit.