In this paper, we develop a new channel model, which we name the $q$-ary partial erasure channel (QPEC). The QPEC has a $q$-ary input, and its output is either the input symbol or a set of $M$ ($2 \le M \le q$) symbols, containing the input symbol. This channel serves as a generalization to the binary erasure channel, and mimics situations when a symbol output from the channel is known only partially, that is, the output symbol contains some ambiguity, but is not fully erased. This type of channel is motivated by non-volatile memory multi-level read channels. In such channels the readout is obtained by a sequence of current/voltage measurements, which may terminate with partial knowledge of the stored level. Our investigation is concentrated on the performance of low-density parity-check (LDPC) codes when used over this channel, thanks to their low decoding complexity using belief propagation. We provide the exact QPEC density-evolution equations that govern the decoding process, and suggest a cardinality-based approximation as a proxy. We then provide several bounds and approximations on the proxy density evolutions, and verify their tightness through numerical experiments. Finally, we provide tools for the practical design of LDPC codes for use over the QPEC.