This paper investigates the impact of puncturing on finite-length polar codes in which a puncturing pattern $\pv^{N}=(p_0,...,p_N)$ is applied to a length-$N$ polar code.. We first introduce two virtual channels to stochastically model the punctured (untransmitted) bits, which are respectively called {\em useless channel model} (UCM) and {\em deterministic channel model} (DCM). Under each model, we derive boolean functions in variables $p_0,...,p_{N-1}$ that can indicate which polarized channels should carry frozen bits. Based on this, we present an efficient method to jointly optimize a puncturing pattern and an information set. Focusing on a fixed information set, we show that there exist the so-called {\em catastrophic} puncturing patterns that will surely lead to a block error and derive their weight distributions recursively. We then propose the two construction methods of a rate-compatible (RC) polar code which ensures that each puncturing pattern in the family is non-catastrophic. Simulation results demonstrate that the proposed RC polar code outperform the RC Turbo code adopted in LTE.